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.OTZE'S SYSTEM OF PHILOSOPHY

PART I

LOGIC

HENRY FROWDE

OXFORD UNIVERSITY PRESS WAREHOUSE
AMEN CORNER, E,C.

LOGIC

IN THREE BOOKS

OP THOUGHT, OF INVESTIGATION
AND OF KNOWLEDGE

ISY

HERMANN LOTZE

ENGLISH TRANSLATION, EDITED

BERNARD BOSANQUET, M.A

FORMIWY FKUOW OF \ NIVFK^ITY ( (>M RGF, OXFORD

Second Edition, in Two Volumes

VOL. I

AT THE CLARENDON PRESS

1888

\_All righh iwnvJ]

EI$TOR'S PREFACE.

SINCE the present Translation of Professor Lotze's
* System der Philosophic ' was begun, both the author him-
self, w^o cordially welcomed the undertaking, and Pro-
fessor Green, who first definitely proposed it, have been
removed by death. These two distinguished men, however
different in method and style of thought, had some funda-
mental tendencies in common ; and it may be of interest to
Professor Lotze's admirers in this country to know that
Professor Green not only executed an important part of the
Translation 1 , but intended to take upon himself the task of
revising and editing the whole 2 , which was not entrusted to
the present Editor till after Professor Green's death.

The Translation of the Logic has been throughout
adapted to the second edition. But the Author's intended
revision of the Metaphysic was not carried out, and the
projected Part III of the * System der Philosophic 7 was
never written. What the Author made known of his
intentions in these respects is mentioned in the Prefaces to
the Metaphysic.

The translation of Part I, the Logic, has been executed
by several hands; the whole of Book I by Mr. R. L.
Nettleship, Fellow of Balliol College, Oxford; Book II,

1 See Preface to the * Metaphysic.'

2 He said to the present Editor : * The time which one spent on such
a book as that (the " Metaphysic") would not be Wasted as regards
one's own work.'

vi EDITOR'S PREFACE.

Chapters i-v (inclusive), by Mr. F. H. Peters, Fellow of
University College, Oxford (with the exception of the ' Note
on the j^ogical Calculus/ which was translated by the
Editor) ; chapters vi-ix (inclusive), by Mr. F. C. Conybeare,
Fellow of University College ; and chapter x by the Editor ;
and the whole of Book III by Mr. R. G. Tatton, Fellow of
Balliol College.

The Editor has revised the whole translation, and is
responsible in all cases for the rendering finally adopted.
He has to thank Mr. J. C. Wilson, Fellow of Oriel College,
Oxford, for the most cordial and ample assistance in deal-
ing with the numerous passages in which mathematical
knowledge was required. It is believed that the translation
of these passages will, owing mainly to his help, be found
on the whole correct and intelligible.

The Table of Contents was furnished by the several
translators for their respective portions. It should be
observed that the original Table of Contents supplies a few
headings (in Book I only), besides those of the chapters.
These are distinguished from the headings supplied by the
translator by being printed in italics. The Index was added
by the Editor.

No endeavour has been made to introduce uniformity of
style into the different portions of the translation. But in
the case of a few important technical terms it has been
thought advisable to introduce renderings as nearly uniform
as the context would allow. Unavoidable variations in the
translation of a German word, or ambiguities in the
employment of an English one, are pointed out, to some
extent, in the Notes and Index ; and in all cases references
are freely give^ to any passages that explain the precise
point of the Author's choice of words. It is hoped that by

EDITOR'S PREFACE. vii

this means the reader may be assisted to master the some
what subtle distinctions which govern the Author's usage,
without the aid of a Glossary, which could indicate them
but roughly. Still Professor Wallace's observations on the
meaning of some German terms, prefixed to his translation
of Hegel's shorter Logic, will be found useful by many
readers.

Jn the case of two of the sections which treat of mathe-
matical questions (234 and 237) the Editor found himself
in a perplexity which could have been removed if the
Author had been still living. The reasoning of sect. 234
seemed more than doubtful ; while the Author himself had
requested the suppression of sect. 237 as 'wholly errone-
ous,' regretting that he had put forward such ( nonsense,'
and explaining that he had been ' misled by the error of a
text-book.'

This unqualified condemnation seemed on consideration
hardly to apply to sect. 237, and to be such as might have
been intended for sect. 234 ; but as the Author mentioned
not only the number of sect. 237, but the pages on which
it stands, the hypothesis of a mere clerical error is almost
excluded. It is nevertheless conceivable that there may
have been some misapprehension ; and therefore it has been
thought advisable not to withdraw sect. 237 entirely, but to
print it as an Appendix.

In preparing the second English edition of the 'System
der Philosophic ' no important alterations have been made
in the translation, although a few verbal corrections have
been found necessary.

AUTHOR'S PREFACE.

THOUGH I venture to describe the present work as the
first part of a System of Philosophy, I hope that this desig-
nation will not be supposed to indicate the same pretensions
which it was wont to herald in times gone by. It is obvious
that I can propose to myself nothing more than to set forth
the entirety of my personal convictions in a systematic
form ; such a form as will enable the reader to judge not
only to what degree they are consistent with themselves,
but also how far they are capable of serving to unite the
isolated provinces of our certain knowledge, in spite of the
great gaps that lie between them, into a coherent view of
the world bearing the character of completeness. In the
present volume, which begins my exposition, I have been
guided, as I shall be in the others, by this purpose.

In the First Book, although entirely rewritten, I have
followed in essentials the line of thought of my short work
on Logic of 1843, which has long been out of print. I
have not seen reason to depart from this line, to which my
own interest in the exposition of Logic is as much confined
now as it was then. Now, as then, I consider it useless
labour to attempt extensions and improvements of the
formal part of Logic, within the limits of the general
character which in fact and' of necessity attaches to it ; but
whatever in it appeared worth knowing, if only as belong-
ing in a certain sense to the history of culture, I have to
the best of my belief conveyed without omission, and have
taken pains to do so as simply as possible.

The Second Book needs no preface ; it is quite free

X AUTHOR'S PREFACE.

ftopci the bonds of system, and simply puts together what-
ever I thought useful. The selection of matter might be
different in many parts, a great deal might be adcted, and
a great deal, it will be thought, might be spared. The
reader should regard it as an open market, where he may
simply pass by the goods he does not want.

The original purpose of the Third Book has not been
carried out. It was meant to treat of the subjects with
which it does in fact deal, on the method of a historico-
critical exposition of systematic logical views the .views
which have appeared both in Germany and in several other
countries in a variety of forms that demand a high degree
of interest and appreciation. But it became clear on
making the attempt that such a task could not be achieved
within the limits of the present treatise, not, that is to
say, with the thoroughness due to the valuable works in
question. Another opportunity may possibly be found for
it ; but in the meantime I was induced by the failure of
this plan to dispense for the present with all reference to doc-
trines which are not my own, and to put forward nothing
but what either is common property, or belongs to my own
individual mode of viewing the subject. I trust that the
whole of my doctrine is not merely of this latter kind !
GOTTINGEN : Jttnc JO, 1874.

The present (2nd) edition contains a number of improve-
ments in detail, and a single addition of some length, the
c Note on the Logical Calculus/ p. 208 (E. Tr. vol. i. p. 275).
I may remark with reference to p. 222 (E. Tr. vol. i. p. 297)
that Jevons speaks of Potassium. Perhaps the reader can
conjecture why J have preferred to speak of Sodium.

GOTTINGEN: September 6, 1880.

\

TABLE OF CONTENTS.

Of Thought (Pure Logic).
INTRODUCTION.

PAGE

Section I. Coherent and coincident ideas ..... i

,, If. Necessary connexion of ideas generally ... 2

,, III. Connexion of ideas established by thought ... 3

,, IV. Is thought an activity rather than an event? . . 3
V. Or, if an activity, has it any specific function in giving

truth? 4

,, VI. Thought in man compared with the other animals . 4
,, VII. Thought adds the notion of coherence to connected

ideas ......... 6

,, VIII. Have the forms of thought a real validity? ... 7
IX. Provibional answer, assuming thought to be a means

to knowledge ........ '8

,, X. Logical distinguished fiom Psychological enquiries . 10
,, XI. Order of enquiry. Pure or formal logic . . .10

XII. Applied logic 1 1

XIII. Knowledge . . . . . . . .12

CHAPTER I.

THE THEORY OF THE CONCEPT.

A. The formation of impressions into ideas.

1. Conversion of impressions into ideas the first work of thought . 13

2. This is effected by the logical act of naming .... 14

3. Which may be called an act of objectification . . 14

4. In the specific forms of the parts of speech . . . .16

5. Relation of substantive, verb, adjective, to substance, event,

property . . . . . . . . 17

6. Relation of thought to its linguistic expression . . 19

7. The other parts of speech. Prepositions and conjunctions . 20

8. Judgment not rightly treated in pure logic before conception . 2 a

xii TABLE OF CONTENTS. [VOL.

PACK

B. Position, distinction, and comparison of the matter of simple ideas.

9. Further r reaction of thought in positing, distinguishing, and

comparing ideas ........ 24

10. Position ' ; practically inseparable from ' objectifi cation *

.. ..

11. Position implies and renders possible distinction ... 26

12. Comparison ; supposed to involve falsification of impressions

by generalisation . . . . . . . 2 7

13. Such a view ignores the true logical ixupoit of the act . . 28

14. Comparison implies a 'universal'; this not a universal 'con-

cept' .......... 30

15. Like all universals, it is not strictly an ' idea ' . . 31

16. Distinction of instances of a universal implies idea of more

and less .......... 32

17. Also those of unity and multiplicity, and greatness and small-

ness .......... 33

18. Logic is not concerned with the origin of these ideas, or with

deductions from them ....... 34

19. In the operations of (B) as compared with those of (A) thought

may be called receptive ...... 35

C. The formation of the concept.

20. Synthesis in consciousness : its various forms . . -37

21. The form of ' conception ' implies the idea of a ground 'for the

synthesis ......... 38

22. Comparison of different instances and observation of the same

instance under different circumstances .... 40

23. * Abstraction ' involved in comparison does not mean mere

omission of differences ....... 41

24. The formation of the second universal (logical concept) implies

the first (14) ........ 42

25. Terminology : ' content/ ' extent,' ' co-ordination,' * subordina-

tion,' * subsumption ' ...... 44

26. Concept ' is rightly applied to individual things ... 45

27. Universals not necessarily concepts; they may be general

images .......... 46

28. Marks not merely co-ordinated, but also mutually determined,

in a concept ......... 47

29. Subordination of species to genus, and subsumption of species

and genus under a mark ....... 49

30. Species = a universal which can be imaged, genus = one

which can only be formulated ..... 50

I.I TABLE OF CONTENTS. xiii

pxia

31. Inverse ratio of content and extent, how far true or important "* 51

32. Kinds of marks : ' differentia,' ' property,' ' accident * . . 53

33. Idea of a complete system of concepts . . v -55

Transition to the Form of the Judgment.

34. Change makes such an idea unrealisable, and leads from con-

ception to judgment 56

35. Conception itself involves questions which also lead to judg-

ment 57

CHAPTER II.

THE THEORY OF THE JUDGMENT.

Preliminary observations on the meaning and customary division
of judgments.

36. Judgment expresses a relation between the matter of two ideas 59

37. This is indicated in the terms Subject, Predicate, and Copula 60

38. Judgments differ according to the different senses of the copula 61

39. The logical sense of the copula is not affected by the quantity

of the judgment 62

40. Nor by its quality ......... 63

41. Nor by its modality ', as ordinarily understood ... 64

42. True apodeictic modality is found in the three forms of relation 66

43. Not of course that any form of judgment can guarantee its

material truth 67

44. So-called problematic judgments are not truly problematic,

nor are questions or prayers ...... 68

45. The only true problematic modality is expressed by particular

and singular judgments ....... 70

46. The ordinary classification omits or confuses many modal

distinctions ......... 70

THE SERIES OF THE FORMS OF JUDGMENT.

A. The imperson il judgment ; the categorical judgment\;
the principle of identity.

47. The categorical judgment is logically preceded by the im-

personal ......... 72

48. Which does not express mere perception, but implies logical

activity .......... 73

49. Relating a present perception to a permanent though un-

expressed subject ' . -74

50. Difficulty as to the logical import of the categorical judgment 75

xiv TABLE OF CONTENTS. [Vot.

r

PACK

51. It does not mean the identity of the subject and predicate . 75

52. Nor is it explained by saying that the one is predicated of t the

othr 76

53. Nor by reference to the metaphysical relation of substance

and attribute ......... 78

54. In fact, the judgment is indefensible against the principle of

identity .......... 79

55. The logical interpretation of that principle .... 80

B. The particular judgment ; the hypothetical judgment ;
the principle of sufficient reason.

56. The difficulty applies to analytical as well as to synthetical

judgments . . . . . . . . .81

57. The justification of categorical judgments is that they are

really identical . . . . . . . 83

58. Illustrations 83

69. But in that case they are T\Q\. judgments at all in the real sense 86

60. This dilemma is met in the hypothetical judgment by making

the identity conditional . . . . . . .88

61. The idea of a condition implies the assumption of a general

coherence of things ....... 89

62. Logical possibility and meaning of such coherence : cause and

reason .......... 90

63. True formulation of the ' principle of sufficient reason ' . . 91

64. The principle of identity alone is no source of knowledge . 93

65. The princ. rat. stiff, not a necessity of thought, but a fact of

all mental experience ....... 94

66. Responsiveness of thinkable matter to thought illustrated from

19 9 6

C. The general judgment ; the disjunctive judgment ; the dictum de
omni et nullo and the principium exclusi medii.

67. The connexion between reason and consequence must be uni-

versal .......... 96

68. This universality is expressed in the ' general ' judgment . 97

69. Further determination of the predicate in the disjunctive

judgment 99

70. True formulation of dictum de omni et nullo . , . .100

71. Princ. excl. med. is only one case of the ' law of disjunctive

thought* 101

72. Its true logical formulation . . . . . . .102

I.] 7 ABLE OF CONTENTS. XV

PAG** 1

73. Incompatibility of contrary, and compatibility of disparate, ^

predicates . . . . . . . . 103

74. The disjunctive judgment leads on to inference . j .105

Appendix on immediate inferences.

75. Inference ad subalternatam . . . . . . .106

76. Ad subalternantem . . . . , . . .107

77. Ad contradictor iam . . . . . . . .108

78. Ad contrariam and ad subcontrariam . . . . .108

79. Inference by conversion . . . . . . .109

80. Conversion of universal judgments . . . . .110

81. Conversion of particular judgments . . . . . in

82. Conversion by contraposition 112

CHAPTER III.

THE THEORY OF INFERENCE AND THE SYSTEMATIC FORMS.
Preliminary remarks upon the Aristotelian doctrine of Syllogism.

83. Formation of the four figures 114

84. General conditions of valid inference in them . . .115

85. Special conditions in each figure. The first figure . .116

86. The second figure . . . . . . . 117

87. The third figure, when both premises are affirmative . .118

88. The third figure, when the premises are mixed . . .118

89. The third figure, when both premises are negative . .119

90. The fourth figure is superfluous 120

91. Superiority of the first to the other figures . . . .121

92. Reduction of the other figures to the fiist . . . .122

93. Syllogisms with hypothetical premises involve no new

principle . . . . . . . . .123

94. Difference of the relation between reason and consequence from

that between cause and effect . . . . . -125

95. Syllogisms with disjunctive, copulative, or remotive premises 1 26

96. Chains of inference 127

A. Syllogistic inferences ; inference by subsumption ; inference by
induction ; inference by analogy.

97. The Aristotelian or subsumptive syllogisms merely make

explicit what is implied in the disjunctive judgment . 128

98. Such inference by subsumption involves a double circle . 129

xvi TABLE OF CONTENTS. [VOL.

PACK

9 9. Illustrations when the premises are (i) analytical, (2) syn-
thetical 13

100. It must be possible to establish (i) major and (2) minor

premises without full knowledge *3 2

101. Inductive inference as solution of the first requirement . .133

102. The defect of induction (as of subsumption) lies in the

practice, not the principle, of it 135

103. Inference by analogy as solution of the second requirement . 137

104. Defect and justification of analogical inference . . .138

B. Mathematical inferences ; inference by substitution ; inference
by proportion ; constitutive equation.

105. The previous forms of inference deal only with universal

subjects and predicates . . . . . . .140

106. Thus do not satisfy the needs of real thinking, which requires

them to be specific 141

107. They are in fact inferences from the extent, instead of from

the content, of concepts . . . . . . .142

108. Inference from content, though implying experience, yields

results for logic . . . . . . . .143

109. It implies substitution of an analysed for an unanalysed

middle term 143

110. Remarks on the symbolisation of logical relationships . . 145

111. Inference by substitution is only strictly applicable to pure

quantities . . . . . . . . .146

112. Still, as an ideal of thought in general, it has its place in

logic 147

118. Extension of it to incommensurable objects in the form of

proportion 148

114. Illustration from Geometry 150

115. Limitation of inference by proportion. Ultimate disparity of

things 151

116. Proportion between marks is modified by the constitution of

the whole subject 153

117. Inference by proportion thus leads to the idea of constitutive

concepts . .154

118. Which however are only fruitful in Mathematics, where all

is commensurable 155

119. To deal with disparate marks, we must go on to classifi-

cation. 157

1.J TABLE OF CONTENTS. xvii

C. The systematic forms : classification ; explanatory theory ; ,
the dialectic ideal of thought.

j PAGE

120. ' Concept '= not a mere sum of marks, but a sum connected

according to a rule .157

121. Many such concepts are formed unconsciously by 'psychical

mechanism' ......... 158

122. Hence the idea that any group of common inarms forms a

concept . . I59

123. Whereas the true concept is found only in the union of essen-

tial marks ......... 160

124. The distinction of essential from unessential leads to com-

parison and classification . . . . . .161

125. Artificial or ' combmatory ' classification .... 162

126. Its defects. It may include more than the facts . . .164

127. Or less than the facts 164

128. And it takes no account of the different values of marks m

the concept 165

129. Logical classification aims at ' constitutive ' concepts, or

'ideas' . . . . . . . . . .167

130. Which naturally connect with the notions of active tendency

and purpose ......... 168

131. And so with that of more and less perfect species . . 1 70

132. Illustrations from Mathematics of the gradation of species . 172

133. The logically most perfect species is that in which all the

marks are at the highest value allowed by the genus . 1 74

134. But each genus may itself have its standard of perfection in a

higher genus . . . . . . . . .175

135. And so lead ultimately to a highest genus which governs the

development of all the rest . . . . . . 177

136. Thus we get the ideal form of natural classification . .178

137. Stationary and progressive perfection of species. ' Type '

and 'Ideal' . . . . . . . . . 179

138. The form of classification by development (like other logical

forms) is only an ideal . . . . . . . 1 80

139. The development of concepts is conditioned by something

other than the concepts themselves . . . . .181

140. This condition must find a place in a complete logical

system . . . . . . . . . .182

141. No theory of ' emanation ' of one concept from another can

dispense with it 183

142. And indeed classification itself leads beyond the particular

concept to universal laws of connexion between its marks 185
LOGIC, VOL. I. b

xviii TABLE OF CONTENTS. [VOL.

PACK

K3. This does not contradict, but confirms, the previous repre-
sentation of the concept as determining the connexion ot
it% marks . . . . . . . . .186

144. A thing is the result, according to universal laws, of the sum

of its conditions ........ 187

145. This view dominates modern science, which explains, in con-

trast with ancient, which classifies. Mechanical character

of the former . . . , . . . . .188

146. Unsatisfactoriness of its ideal 190

147. Antagonism between scsthetic and scientific view of things.

Possible reconciliation . . . . . . .191

148. 'Laws' are not external to reality, but constitute its very

nature .......... 193

149. Form of the ultimate ideal of thought . . . . 194

150. Supposed analogy of the living organism. Hegelianism.

1 Speculation ' ........ 195

151. Value of the ' speculative * ideal. It belongs to logic, but

points beyond it . . . . . . 197

BOOK II.

Applied Logic.
152-3. Prefatory Remarks . . . . . . . . 199

CHAPTER I.

THE FORMS OF DEFINITION.

154. Ideas, how communicable ....... 20,2

155. Poetry and rhetoric 203

156. Uncertainty of communication ...... 204

157- Explanation by abstraction ...... 205

158. This the only method for simple ideas . . . .206

159. Explanation by construction. Description .... 207
160-1. Description and definition ...... 209

162. Nominal and real definitions . . . . . -213

163. Three faults to avoid 214

164. Elegance and brevity . . . . . . . .216

165. Evil of superfluity 218

166. Popular definitions . . . . . . . .219

167. Genetic definition . . . . . . . .220

168. The end of definition is the conception , . .222

TABLE OF CONTENTS. xix

CHAPTER II.

OF THE LIMITATION OF CONCEPTIONS.

PA(,E

169-70, We must start from the conceptions already expressed in

language 225

171. Disparate groups of sensations 227

372. Popular language justified ....... 229

173. Relations between the members of these groups. Tastes.

Colours . . . . . . . . . -231

174. Scale of sounds . . . . . . . . -233

175. Heat-sensations . . . . . . . . -235

176. Arbitrariness of scale ........ 236

177. Illustrations from practical life . . . . . -237

178. Moral and aesthetic distinctions ...... 239

179. Transition from concave to convex ..... 241

180. The distinctions remain in spite of the transition from one

conception to another . . . . . . .242

181. And though there be a term in the series that satisfies both

conceptions ......... 243

182. Illustrations ......... 245

183. Development ......... 246

CHAPTER III.

SCHEMES AND SYMBOLS.

^84. The notion of a universal scheme or system of conceptions . 249

185. Pythagorean ism . . . . . . . . .250

186. Grandeur of its general idea ...... 252

187. Poverty of the particular form in which it is expressed . 254
388. Numbers and things 255

189. Other kindred speculations 257

190. Demand for symmetry . . . . . . .259

191-5. The Hegelian dialectic 262

196. The scheme of Leibnitz 271

197. Is such a scheme possible ? 273

198. What it would require 275

Note on the Logical Calculus . . . . . a 77

b 2

xx TABLE OF CONTENTS. [VOLS.

CHAPTER IV.

THE FORMS OF PROOF.

PAGF.

199. Discovery and proof. Proof of particular and of universal

propositions . . . . . . . . . 299

200. Proof rests on axioms. Axioms how distinguished . . 30 r

201. Before starting to prove a proposition we must know that it

is worth proving, i. e. that (a] the ideas are definite . 303

202. () their combination possible ...... 304

203. (Y) the proposition true in fact 306

204. Eight forms of proof distinguished 307

205-6. (i) First direct progressive proof from the conditions of*?""

to r 308

207. (2) Second direct progressive proof from T to its conse-

quences 312

208. (3) First direct regressive proof from Tto its conditions . 313
209-10. (4) Second direct regressive proof from the consequences

of Tto T 315

211. (5) First indirect progressive proof . . . . . 317

212. (6) Second indirect progressive proof . . . . . 319

213. Indirect regressive proofs 321

214-5. What is ordinarily called proof by analogy is really proof

by subsumption . . . . . . . -322

216. The mathematician's proof by strict analogy is also proof by

subsumption . . . . . . . . . 328

217. Analogy and the Dictum de omni et nullo .... 330

CHAPTER V.

THE DISCOVERY OF GROUNDS OF PROOF.

218. No rules for the discovery of a proof, but the problem itself

may give a clue 333

219. Illustiations from Geometry ...... 334

220-5. The conditions of equilibrium ...... 336

226-7. The principle of the lever 343

228-9. Rotatory motion 347

230. A line without mass cannot be moved . . . . .352

231-5. The parallelogram of forces ...... 354

236. Difficulty of analysis 364

237. Suppressed.

238-9. The Taylorian theorem . . . . . .367

r. n.]. TABLE OF CONTENTS. xxi

CHAPTER VI. \Vol.IL

FALLACIES AND DILEMMAS.

PACK

240. Premisses must be true in order to prove a conclusion . . i

241. And must not covertly involve the conclusion ... 2

242. Preposterous Reasoning confuses the principiatum as causa

cognoscendi with the principium as causa essendi . . 3

243. Ambiguity of middle term mostly due to the confusion of a

relative with an absolute truth ..... 4

244. Illustration of the above from moral precepts, all of which

have their exceptions 5

245. As have also mechanical formulae, which become unmeaning,

when pushed to extremes ...... 7

246. Fallacies of too wide or too narrow definition . . .10

247. Fallacy of incomplete explanation illustrated by the popular

idea that lapse of time destroys motion . . . .10

248. Incomplete disjunction the cause of much philosophical and

other onesidedness . . . . . . . .12

249. The fallacy in Zeno's paradoxes about reality of motion . 14

250. Examples of classical dilemmas stated and explained . . 17

CHAPTER VII.

UNIVERSAL PROPOSITIONS AS DERIVED FROM PERCEPTIONS.

251. Inductive methods are based on results of deductive Logic . 22

252. Connexions of elements revealed in sensible experience are

mostly impure ........ 23

253. The universality of a pure connexion or its character as a

law of nature guaranteed by the law of Identity . . 24

254. The raw matter of Inductions consists not of passive im-

pressions but of perceptions already articulated by
thought as subject and predicate and ranged under
general conceptions . . . . . . .25

255. They are so ranged by an incomplete analogy, based on a

distinction of essential from non-essential remarks, which
logical theory cannot assist . . . . . .28

256. In reaching universal inductions we must argue ad subaltern-

antem 31

257. The truths of Geometry are universal because the diagram is

used as a symbol only of our conception . . 33

258. The highest inductions not categorical but hypothetical

judgments 35

xxii TABLE OF CONTENTS. [VOL.

PAGE

1>$9. Terms which are exclusively cause and effect of each other

are related as ground and consequent . . . -37

260. Experiment merely subsidiary to observation and rft,s no

peculiar virtue of its own 38

261. Typical cases of the relation in which two phenomena C and

E may stand to one another ...... 40

(t) C and E Co-Exist always.

(2) C and E frequently concur.

(3) Absence of C not involving absence of E. Criticism

of the canon ' cessante causa cessat et effectus.'

(4) Presence of C not involving presence of E. Differ-

ence of relation of cause and effect and of ground
and consequent.

(5) Absence of C involving absence of E.

(6) Presence of E involving presence of C. Criticism of

Newton's canon ' effectuum naturalium ejusdem
generis eaedem sunt causae.'

(7) Absence of E involving absence of C.

262. Whether the phenomenon C is or only contains the cause of

E can only be decided by analysis of both into their
elements and observation of which elements of the one
involve which elements of the other. Typical examples
of such analysis . . . . . . . -52

263. The exact nature of the causal nexus inferred from any of

the above relations to exist between C and E can only be
apprehended by observation of the quantitative changes
they cause in one another. Examples of such quantita-
tive correspondences ....... 60

CHAPTER VIII.

THE DISCOVERY OF LAWS.

264. Science not content with discovering a mere connexion

between two phenomena seeks to know the law of this
connexion ......... 67

265. Laws of nature are universal hypothetical judgments and not

assertions of universal matters-of-fact .... 68

266. A law expresses an objective and intelligible connexion of

phenomena, a rule is a mere subjective method of thought 71

267. The ultimate criterion of sense-perception to be found in

sense itself 73

268. Facts as they appear are not only relative to one another

n 1 TABLE OF CONTENTS. xxiii

PAGF

but to the standpoint of the observer, and must therefore

be grasped as projections of ulterior and truer facts 76

269. A lav always transcends the given, being an extension to

cases not given of what holds good within the given '. A
truly universal law is not a demonstrable truth . . 79

270. Laws based on statistics are mostly partial truths . . 84

271. The law which prima facie best fits in with observed facts

need not therefore be the truest expression of their inter-
connexion ......... 85

272. Simplicity no guarantee for the truth of a law. The simplest

law only preferable where it is the sole conceivable one . 87

273. A postulate lays down the conditions under which alone the

.given appearance is conceivable. An hypothesis is a
suggestion of conceivable facts fulfilling the demands of
the postulate and so explaining the appearance. A fiction
views the given as an approximate realisation of a known
law, in the absence of a known law to which it can be
simply referred 90

274. Rules for framing of hypotheses not to be laid down before-

hand, but none to be rejected because beyond reach of
refutation if false ........ 94

275. Hypotheses must satisfy their postulates and supply the

conditions of the appearances to be explained ... 97

276. An old hypothesis not to be hastily set aside but modified to

suit the new and discrepant facts ..... 99

277. An hypothesis must limit itself to asserting what is possible,

i.e. what can be conceived or pictured as matter-of-fact . 101

CHAPTER IX.

DETERMINATION OF INDIVIDUAL FACTS.

278. In determining facts which transcend the immediate im-

pression we must be guided by probability . . .104

279. In view of the complexity of things a principle of ex-

planation must not be too simple and abstract . .105

280. And on the other hand it must involve as few presuppositions

as possible. Positive evidence preferable to negative . 107

281. The mathematical determination of chances assumes that

they are all equally possible, but that one of them must
occur 109

xxiv TABLE OF CONTENTS. [VOL.

PACK

2if2. (i) Mathematical chance no positive prediction of events.

It measures our expectation of their occurrence - . 113

(2) Jheory of composite chances.

(3) dependent chances.

(4) probability of alternative causes.

(5) probability of an event's recurrence.

(6) mathematical expectation.

(7) moral expectation.

283. Calculus of chances not only presupposes the laws of all

calculation such as law of Identity and doctrine of dis-
junctive judgment, but also an ordered universe of inter-
dependent events 127

284. Mathematical chance is our subjective expectation of vin

event, and not a permanent property thereof. The
resulting chance improbable only as compared with the
sum of its alternatives, not as compared with any one of
them . . . . . . . . . .130

285. Success of attempts made to test by experiment the calculus

of chances 133

286. Such successful results not fraught with intelligible necessity,

but the result of constant conditions operating among
vaiiable ones, which in the long run neutralise each other 135

287. Use of the calculus in cases where constant and variable

causes of an often repeated event are unknown. Nature

of so-called statistical laws . . . . . .139

288. Use of the calculus in determining the probable accuracy of

our observations of magnitude. The method of the least
squares .......... 142

CHAPTER X.

OF ELECTIONS AND VOTING.

289. Conditions presupposed by a logical treatment of the problem

of expressing a collective will ...... i

290. Defects of absolute majority . . . . . 149

291. The weight of votes. A majority of majorities may be a

minority of the whole constituency . . , 149

292. Voting so as to express intensities of Volition . . .153

293. Election by elimination 157

294. When order of putting proposals to the vote is important . 159

295. Rejection of innovations as stick. ' Order of the day . .161

296. Amendments and substantive motion. Order of putting

proposals to the vote . . . . . . .162

H.f TABLE OF CONTENTS. xxv

BOOK III.

On Knowledge (Methodology).
INTRODUCTION.

PACK

2!) 7. Analytic and Synthetic methods practically inseparable 166

298. Correspond respectively to Investigation and Exposition ; are

more general than ' methods ' of applied Logic . . 1 69
21)9. But applied Logic, like common thought, rests on untested

bases J 7

300. And so does science as we have it . . . T 7 T

301. Methodology however as treatment of Knowledge is enquiry

into sources of certainty . . . . . *73

CHAPTER I.

ON SCEPTICISM.

302. Scepticism presupposes Truth and Knowledge . . .176

303. But doubts whether our Knowledge is Truth. Descartes . 179

304. This doubt involves the assumption of a world of things

which our thought should copy 182

305. But any decision postulates the competence of thought . 184

306. Which can only be guided by conceptions in our minds . 185

307. Our delusion could only be revealed by fresh knowledge . 187

308. Which must be related to the old. Things are not know-

ledge of things . . . .189

309. That Things may not be what they seem, as a mere general

doubt, is self-contradictory *9 2

310. Sceptical arguments in Sextus Empiiicus . . . . *93

311. They involve the above difficulties . . . * .196

312. Error in ' we only know phenomena ' *9 8

CHAPTER IL

THE WORLD OF IDEAS.

313. Genesis of Plato's doctrine ot ' Ideas ' aoo

314. The Ideas as Universal conceptions 202

315. Possible knowledge of Ideas apart from question of Things . 204

316. Distinction between Existence, Occurrence, Validity . . 206

317. Confusion of Existence and Validity in case of the Ideas . 210

318. Ideas in what sense eternal, And independent of things . 211

xxvi TABLE OF CONTENTS. [VOL.

J-AGE

KJ19. Aristotle on the Ideas. His universal too is ovaia . .214
328. Modern counterparts of the Ideas. Validity a difficult

notion . . . . . . . . . .216

321. * Ideas impart no motion' criticised; importance of Judg-
ments 218

CHAPTER III.

THE A PRIORI AND THE EMPIRICAL METHODS.

322. Judging of knowledge by our notions of its origin an illusion 223

323. Attempt to find a starting-point for knowledge. 'Cogito,

ergo sum' % . 226

324. Innate Ideas; but are they true ? . . . . .229

325. Action of one thing on another implies Spontaneity in order

to Receptivity 231

326. Nature of mind is contributory in #// elements of knowledge 232

327. Both in simple Perception and in such ideas as that of causal

connexion 234

328. External reality must be criticised on ground of knowledge . 236

329. Universality and Necessity as marks of a priori knowledge . 239

330. Universal validity not derivable from repeated perceptions

alone 241

331. There may be spurious self-evidence* which is tested by

thinking the contradictory 243

332. Use of psychological analysis in establishing first principles 246

333. Even modern Psychology hardly helps Logic . . . 248

CHAPTER IV.

REAL AND FORMAL SIGNIFICANCE OF LOGICAL ACTS.

334. Thought must have some Real significance . . . -252

335. Comparison and distinction as acts resulting in Relations . 254

336. Thought is symbolic and discursive . . . . -256

337. How can a relation of ideas be objective .... 259

338. Only as independent of individual mind. The case of Things 260

339. A universal cannot be realised, but has objective validity . 264

340. Nominalism and Realism confuse Existence and Validity . 267

341. The Reality of general notions is only validity . . . 268
342 Conception not akin to object in structure, but in net result . 270

343. Degrees of subjectivity in kinds of Judgment . . . 273

344. Subjective character of Syllogism and Induction . . .276

345. Terms antithetic to ' Subjective* and ' Formal ' . . . 279

II.] TABLE OF CONTENTS. xxvii

CHAPTER V.

THE A PRIORI TRUTHS.

PACK

346. The world of Knowledge and the world of Things . . 283

347. ' Actual Reality ' ; adequacy of Judgments to it . . .286
J48. Applicability of thought to the course of events involves

(i) Some given reality, which thought cannot create . 288

349. (2) The Universality of Law in the Real world ; ultimately a

matter of faith ........ 290

350. And (3) synthetic judgments a priori t as basis of knowledge

of particular laws . . . . . . . .294

351. Hume's restriction of judgment destroys #// judgment . . 295

352. Mathematical reasoning is not covered by the Law of Identity 297

353. Illustration by Kant's arithmetical instance . . . .299

354. And by his geometrical instance ..... 303

355. Meaning and value of apprehension a priori . . . 305

356. Self-evidence of universal Truths ..... 307

357. Intuition is opposed to discursive thought means immediate

apprehension ......... 309

358. Self-evident Truths require to be discovered by help of

analysis 311

359. Pure Mechanics in what sense a priori . . . .313

360. Gradual formation of pure ideas of Motion and Mass . . 316

361. Mechanical principles, like those of Arithmetic and Geometry,

at once identical and synthetic . . . . .319

362. In higher Mechanics, Proof is one thing, and the Ratio legis

another 323

363. Analytical Knowledge as the ideal, means the simplest

synthetical knowledge 325

364. The simplest ultimate Truth need not be a mere datum of

experience, though it must be Synthetic .... 327

365. A synthetic yet necessary development the supreme goal of

science 329

APPENDIX 331

INDEX . 333

BOOK I.

OF THOUGHT (PURE LOGIC).
INTRODUCTION.

I. AT almost every moment of our waking life our senses
are giving rise to various ideas, simultaneous or immediately
successive. Among these ideas there are many which have
a right thus to meet in our consciousness, because in the
reality from which they spring their occasioning causes
always accompany or follow one another ; others are found
together in us merely because, within the external world to
whose influence we are accessible, their causes were as
a fact simultaneous though not so inwardly connected as to
ensure their similar combination in every recurring instance.
1'his mixture of coherent with merely coincident ideas is
repeated, according to a law which we derive from self-
observation, by the current of memory. As soon as any
idea is revivified in consciousness, it reawakens the others
which have once accompanied or succeeded it, whether the
previous connexion was due to a coherence in the matter of
the ideas, or to the mere simultaneity of otherwise uncon-
nected irritants. It is upon the first fact, the recovery
of what is coherent, that our hope of arriving at knowledge
is based : the second, the ease with which coincident ele-
ments hang together and push one another into conscious-

LOGIC, VOL, I. B

2 INTRODUCTION. [Book I.

ness, is the source of error, beginning with that distraction
wi.ich hinders our thoughts from following the connexion
of thinps. e

II. The ever-changing whole of processes which results
from this peculiarity of our psychical life is what we call
the current of ideas. If it were in our power to observfe
this whole with omniscience, we should discover in every
instance of it, in the sober course of waking thought, in the
dreams of sleep, in the delirium of disease, a necessary
connexion between its members. The application of uni-
versal laws, which hold good of all souls alike % to the
particular conditions which are found to vary in each single
instance, would exhibit the course of these inner processes
in the light of an inevitable result. If we knew the per-
manent characteristics of a single particular soul, if we had
a view of the form and content of its whole current of ideas
up to the present time, then, the moment it had produced
a first and a second idea on occasion of external irritants,
we should be able to predict on the basis of those universal
laws what its third and fourth idea in the next moment
must be. But in any other soul, whose nature, past history,
and present condition were different, the same first and
second idea, developed at this moment by a similar external
irritant, would lead with a similar necessity in the next
moment to an entirely different continuation. An investiga-
tion of the subject would therefore have to recognise that
any given current of ideas was necessary for that particular
soul and under those particular conditions ; but it would
not discover any mode of connexion between ideas which
was universally valid for all souls. And just because,
under their respective conditions, every such series of ideas
hangs together by the same necessity and law as every
other, there would be no ground for making any such
distinction of value as that between truth and untruth,
which would place one group in opposition to all the
rest.

Book l.i INTRODUCTION. 3

III. Universal validity and truth are the two prerogative?
which even ordinary language ascribes and confines . co
those connexions of ideas which thought alone is supposed
to establish. Truth is familiarly defined as the agreement
of ideas and their combinations with their object and its
relations. There may be objections to this form of ex-
pression, which this is not the place to consider ; but it will
be innocuous if we modify it and say, that connexions
of ideas are true when they follow such relations in the
matter of the ideas as are identical for all consciousness,
and not such merely empirical* coincidence of impressions
as takes one form in one consciousness, another in another.
Now our ideas are excited in the first instance by external
influences, and this leads us to regard thought as a reaction
of the mind upon the material supplied by those influences
and by the results of their interaction already referred to.
The thinking mind is not content to receive and acquiesce
in its ideas as they were originally combined by casual
coincidence or as they are recombined in the memory : it
sifts them, and where they have come together merely in
this way it does away with their coexistence : but where
there is a material coherence between them, it not only
leaves them together but combines them anew, this time
however in a form which adds to the fact of their recon-
nexion a consciousness of the ground of their coherence.

IV. I will connect the indispensable explanation of what
I have just said with the elucidation of some obvious
objections. It is not without a purpose, which I admit,
that while I have represented the rest of the current of our
ideas as a series of events, which happen in us and to us
according to universal laws of our nature, I have represented
thought as an activity which our mind exercises. There
have been persons who doubted whether this opposition
has any real significance, either in itself or in relation to
thought; whether everything that we are in the habit of
calling activity is not rather one amongst the events which

B 2

4 INTRODUCTION. [Book I.

pimply take place in us. So wide a question does not of
cburse admit of being decided here : if therefore I hold to
the significance of the opposition, and expressly describe
thought as an activity, this must be regarded as a presupposi-
tion which awaits proof elsewhere, but is at present open
to dispute. It is necessary for the connexion of the whole"
to which I wish this view of thought to serve as an intro-
duction; and it seems to me to be permissible, because,
while it will determine decisively the general colour of my
exposition, it will not alter unnaturally the internal relations
of its subject-matter. .

V. It is more profitable to meet another form of the
same objection, which allows the general validity of the
opposition in question, but holds that there is no occasion
to apply it here. The connexion of the coherent (it is
said), that is to say, Truth, is brought about in the same
way, only not quite so soon, as the erroneous conjunctions
of the casually coincident. The course of things itself
ensures that those events which are inwardly connected
exercise their combined effect upon us with incomparably
greater frequency than those which have no inward bond,
but are variously thrown together by chance. Owing to
this more frequent repetition the connexion of what is
coherent becomes fixed in us, while that of the merely
coincident is loosened and disturbed by its want of urji-
formity. In this way the separation of the coherent from
the incoherent, which we thought it necessary to ascribe to
a special reaction of the mind, is effected by the current of
ideas itself; and thus brutes, like men, acquire the mass of
well-grounded information which regulates the daily life of
both. It would be superfluous to point out that this
account is perfectly correct if it purport to be no more than
a history of the acquisition just mentioned ; but I think it
can be shown that this acquisition is just what neither
characterises nor exhausts the specific work of thought.

VI. There is a common opinion which reserves the

Book I.] INTRODUCTION. 5

faculty of thought to man and denies it to brutes. With
out seriously deciding for or against this view, I will use it
for the convenience of my explanation. In the s$ul of a
brute, which on this theory would be confined to a mere
current of ideas, the first impression of a tree in leaf would
only produce a collective image ; there would be no power
or even impulse to seek for any special coherence between
its parts. Winter strips the tree of its leaves, and on a
second observation the brute finds only a part of the former
collective image, which tries to reproduce the idea of the
rest, bitf is hindered by the present appearance. When the
return of summer restores the old state of things, the
renewed image of the whole tree in leaf may not, it is true,
have the simple unquestioning unity of the first observation ;
the recollection of the second intervenes, and separates it
into the part which remained and the part which changed.
I do not think we can say what precisely would take place
in the soul of the brute under these circumstances ; but
even if we ascribe to it the additional faculty of comparing
and surveying the current of its ideas and expressing the
result, the expression could not say more than the fact that
two observations were at one time together, at another not.
Now it is true that the man, when he gives the name of
leafy and leafless tree to the same observed objects, is only
expressing the same facts ; but the apprehension of the
facts, which is indicated by these habitual forms of speech,
involves a mental operation of quite a different kind. The
name of the tree, to which he adds and from which he
takes away the descriptive epithet, signifies to him, not
merely a permanent as opposed to a changeable part in his
observation, but the thing in its dependence on itself and
in opposition to its property. The effect of bringing the
tree and its leaves under this point of view is, that the
relationship of thing and property appears as the justifica-
tion both for separating and for combining these ideas, and
thus the fact of their coexistence or non-coexistence in our

6 INTRODUCTION. [Book I.

5gnsciousness is referred to the real condition upon which
their coherence or non-coherence at the moment depends.

The rsame consideration may be extended to other
instances. In the soul of the dog the renewed sight of the
raised stick recalls the idea of his previous pain : the man,
when he makes the judgment, ' the blow huFts,' does no
merely express the fact of connexion between the two
occurrences, but justifies it. For in representing the blow
as the subject from which the pain proceeds, he clearly
exhibits the general relationship of cause and effect as the
ground, not of the mere coexistence of the two ideals in us,
but of their right and obligation to follow one another.
Lastly, the expectation of pain in the dog may be accom-
panied by the recollection that by running away, to which he
was led before by an involuntary instinct, the pain is
diminished ; and this fresh conjunction of ideas will doubt-
less make him repeat the salutary operation as surely as if
he reflected and concluded that threatening blows are pre-
vented by distance, that a blow threatens him, and that
therefore he must run away. But the man who in a similar
or more serious case actually frames such a conclusion,
performs an entirely different mental operation ; in express-
ing a universal truth in the major premiss, and bringing a
particular instance under it in the minor, he not merely
repeats the fact of that salutary connexion between ideas
and expectations by which the brute is affected, but he
justifies it by an appeal to the dependence of the particular
upon its universal.

VII. These examples, which embrace the familiar forms
of thought, concept, judgment, and syllogism, will I think
have made sufficiently clear what is the surplus of work
performed by thought over and above the mere current of
ideas ; it always consists in adding to the reproduction or
severance of a connexion in ideas the accessory notion of a
ground for their coherence or non-coherence. The value
of this work remains entirely the same, whatever opinion we

Book I.] INTRODUCTION. 7

may hold of its genesis : if we preferred to regard it, not as
the outcome of a special activity, but only as a finer procKot
of the mere current of ideas operating under favorable
circumstances, we should then confine the name of thought
to that particular stage of development in the current at
which it gives birth to this new achievement. The pecu-
liarity of thought, then, which will govern the whole of our
subsequent exposition, lies, not in the mere correspondence
of our apprehension with fact, but in the production of
those accessory and justificatory notions which condition
the form of our apprehension. We do not deny that, apart
from thought, the mere current of ideas in the brute gives
rise to many useful combinations of impressions, correct
expectations, seasonable reactions ; on the contrary, we
admit that much even of what the man calls his thought is
really nothing but the play of mutually productive ideas.
And yet perhaps there is still some difference here. The
sudden inspirations which enable us to make a decision in
a moment, the rapid survey which arranges a complicated
material in almost less time than would seem sufficient for
the bare observation of its parts, the invention of the artist
which remains unconscious of the grounds by which it is
impelled, all these seem to us to be effects, not of a current
of ideas which has not yet become thought, but of abbre-
viated thought. In the cases where these surprising
operations are successful, they are so because mature
thought has already in other cases developed into full-grown
habits those accessory notions, which bring the impressions
under universal principles of coherence ; and this, like all
other accomplishments which have acquired the ease of a
second nature, has behind it a forgotten time of laborious
practice.

VIII. In the examples which I have employed, the
accessory notions, by which we justify the connexions of
ideas, obviously coincided with certain presuppositions
about the connexions of the real with which we cannot

8 INTRODUCTION. [Book I,

dispense. Without the opposition of things and their
pi ^oerties by which the whole matter of perception is
articulated, without the assumption of a succession of
effects from causes, and without the determining power
of the universal over the particular, we could have no
apprehension whatever of the reality which surrounds us:
From this point of view, then, it seems a self-evident
proposition that the forms of thought and the accessory
notions which give them vitality are immediate copies of
the universal forms of being and its connexions, and this
real validity of thought and its operations has, in fact, been
frequently maintained. The opposite view to this, which
as its exact counterpart we might expect to find, has never
been put forward so unreservedly. To the unprejudiced
mind it is too natural to regard thought as a means of com-
prehending the real, and any interest in the scientific
investigation of its processes is too dependent upon this
presupposition for any one to assert the merely formal
validity of all logical activity in the sense of denying all
relation between it and the nature of being. Those, there-
fore, who have regarded the forms and laws of thought as
being primarily peculiar results of our mental organisation,
have not wholly excluded their correspondence with the
essence of things ; they have only denied the off-hand view
which would make them immediate copies of the forms
of being.

IX. In regard to this much debated question an intro-
duction can only take up a provisional position. We shall
certainly be right to confine our attention at starting to
what is already clear, and to leave for a later stage the
decision of uncertainties. Let us then go no further than
the natural presupposition which regards thought as a means
to knowledge. Now a tool must fulfil two conditions, it
must fit the thing and it must fit the hand. It must fit the
thing ; that is, it must be so constructed as to approach,
reach, and get hold of, the objects which it is to work upon,

Book I.] INTRODUCTION. g

and find in them a point from which to operate; this
requirement is satisfied in the case of thought if we acKat
that its forms and laws are no mere singularities y of our
mental organisation, but that, taken as they are, they show
a constant and regular adaptation to reality. If, again,
a tool is to fit the hand, it must have such other structural
properties as make it easy to grasp, hold, and move, having
regard to the power, attitude, and position of the person
who is to use it ; and in the case of thought this second
indispensable requirement limits the scope of the previous
admissipn. Only a mind which stood at the centre of the
real world, not outside individual things but penetrating
them with its presence, could command such a view of
reality as left nothing to look for, and was therefore the
perfect image of it in its own being and activity. But the
human mind, with which alone we are here concerned, does
not thus stand at the centre of things, but has a modest
position somewhere in the extreme ramifications of reality.
Compelled, as it is, to collect its knowledge piece-meal
by experiences which relate immediately to only a small
fragment of the whole, and thence to advance cautiously to
the apprehension of what lies beyond its horizon, it has
probably to make a number of circuits, which are im-
material to the truth which it is seeking, but to itself in the
search are indispensable. However much, then, we may
presuppose an original reference of the forms of thought to
that nature of things which is the goal of knowledge, we
must be prepared to find in them many elements which do
not directly reproduce the actual reality to the knowledge
of which they are to lead us : indeed there is always the
possibility that a very large part of our efforts of thought
may only be like a scaffolding, which does not belong to the
permanent form of the building which it helped to raise,
but on the contrary must be taken down again to allow the
full view of its result. It is enough to have thus raised
a preliminary expectation, with which we wish our subject

10 INTRODUCTION. [Book I.

to be met; any more definite decision as to the limits
\vHch separate the formal validity of our thought from its
real significance must await the further course of our en-
quiries.

X. I have purposely avoided postponing those enquiries
by discussions which seem to me to encumber unjustifiably
the approach to logic. What particular tone of mind is
required for successful thinking, how the attention is to
be kept up, distraction avoided, torpidity stimulated, pre-
cipitation checked, all these are questions which no more
belong to the field of logic than do enquiries about the
origin of our sense-impressions and the conditions under
which consciousness in general and conscious activity is
possible. We may presuppose the existence of all these
things, of perceptions, ideas, and their connexion according
to the laws of a psychical mechanism, but logic only begins
with the conviction that the matter cannot end here ; the
conviction, that between the combinations of ideas, however
they may have originated, there is a difference of truth and
untruth, and that there are forms to which these combina-
tions ought to answer and laws which they ought to obey.
It is true that we may attempt by a psychological investiga-
tion to explain the origin of this authoritative consciousness
itself; but the only standard by which the correctness
of our results could be measured would be one set up
by the very consciousness to be investigated. The first
thing, then, that has to be ascertained is, what the con-
tents of this authoritative conviction are ; the history of its
growth can only have the second place, and even then must
conform to requirements of its own imposing.

XI. Having now said all that seemed necessary by way
of introduction to my exposition, I will add a preliminary
survey of its order. The examples which we have hitherto
employed lead naturally to a first principal part, which,
under the name of pure or formal logic, is devoted to
thought in general and those universal forms* and principles

Book I.] INTRODUCTION. II

of thought which hold good everywhere, both in judging of
reality and in weighing possibility, irrespective of any differ-
ence in the objects. We have only to mention concept,
judgment, syllogism, to see how naturally these forms
exhibit themselves as different stages of one and the same
activity; and in treating of pure logic I shall endeavour
to emphasise this thread of connexion somewhat more
strongly than is usually done. The various forms of
thought will be arranged in an ascending series, in which
each higher member attempts to make good a defect
in the preceding one, due to its failure to satisfy, in regard
to its own particular problem, the general impulse of
thought to reduce coincidence to coherence. This series
will advance from the simplest formation of single impres-
sions to the conception of the universal order in which
this general impulse would lead us, if it were possible,
to comprehend the world.

XII. Pure logic itself will show and explain that the
forms of concept, judgment, and syllogism are to be con-
sidered primarily as ideal forms, which give to the matter of
our ideas, if we succeed in arranging it under them, its true
logical setting. But the different peculiarities of different
objects offer resistance to this arrangement ; it is not clear
of itself what sum of matter has a claim to form a deter-
minate concept and be opposed to another, or which
predicate belongs universally to which subject, or how the
universal law for the arrangement of a manifold material is
to be discovered. Applied logic is concerned with those
methods of investigation which obviate these defects. It
considers hindrances and the devices by which they may be
overcome ; and it must therefore sacrifice the love of
systematisation to considerations of utility, and select what
the experience of science has so far shown to be important
and fruitful. The boundlessness of the field of observation
unfortunately makes it impossible to exhibit as completely
as could be wished this most brilliant part of logic, which

12 INTRODUCTION.

*he inventive genius of modern times has made peculiarly
its nwn.

XK^ The third part will be devoted to knowledge, that
is, to the question which our introduction touched without
answering, the question how far the most complete structure
of thought which all the means of pure and applied logic
enable us to rear, can claim to be an adequate account
of that which we seem compelled to assume as the object
and occasion oF our ideas. The currency in ordinary
minds of this opposition between the object of our
knowledge and our knowledge of that object makes me
employ it without hesitation to describe in a preliminary
way the subject of this third section ; it may be left to the
section itself to disclose the difficulties which this apparently
simple antithesis involves, and to determine accordingly
the more precise limits of the problems with which it has
to deal.

CHAPTER I.

The Theory of the Concept.
A. The formation of impressions into ideas.

1. IT is in relations within a manifold that the operations
of thought usually show themselves to us, and we might
therefore expect to have to look for the most original of its
acts in some simplest form of connexion between two ideas.
A slight reflexion, however, suggests to us to go a step
further back. It is easy to make a heap out of nothing but
round stones, if it is indifferent how they lie; but if a
structure of regular shape is to be built, the stones must be
already so formed that their surfaces will fit firmly together.
We must expect the same in the case before us. As mere
internal movements, the states which follow external irritants
may exist side by side in us without further preparation, and
act upon each other as the general laws of our psychical life
allow or enjoin. But if they are to admit of combination
in the definite form of a thought, they each require some
previous shaping to make them into logical building-stones
and to convert them from impressions into ideas. Nothing
is really more familiar to us than this first operation of
thought ; the only reason why we usually overlook it is that
in the language which we inherit it is already carried out,
and it seerns therefore to belong to the self-evident pre-
suppositions of thought, not to its own specific work.

14 THE THEORY OF THE CONCEPT. [Book I.

2. That which takes place in us immediately under the
iriVuence of an external stimulus, the sensation or the
feeling is in itself nothing but a state of our consciousness,
a mood of ourselves. We do not always succeed in
naming, and so making communicable to others, the
manner in which we are thus affected; sometimes the
formless interjection, the exclamation, is the only way we
can find, though with no certainty of being understood, to
give sound to what cannot be said. But in the more
favorable cases, where we have succeeded in creating a
name, what exactly is it which this creation effects and
indicates? It is just what we are here looking for, the
conversion of an impression into an idea. As soon as we
give the name of green or red to the different movements
which waves of light produce through our eyes, we have
separated something before unseparated, our sensitive act
from the sensible matter to which it refers. This matter
we now present to ourselves, no longer as a condition
which we undergo, but as a something which has its being
and its meaning in itself, and which continues to be what
it is and to mean what it means whether we are conscious
of it or not. It is easy to see here the necessary beginning
of that activity which we above appropriated to thought as
such : it has not yet got so far as converting coexistence
into coherence, it has first to perform the previous task .of
investing each single impression with an independent
validity, without which the later opposition of their real
coherence to mere coexistence could not be made in any
intelligible sense.

3. We may describe this first operation of thought as the
beginning of an obj edification of the subjective ; and I take
advantage of this expression to guard against a misunder-
standing and so illustrate the simple meaning of what I
have said above. It is not objectivity in the sense of some
sort of real existence which would subsist though nobody
had the thought of it, that, by the logical act of creating a

Chap, I.] LOGICAL OBJECTIFICATION. 15

name, is accorded to the subject-matter to which that act
gives rise. The true meaning of the first act of though/is
best exemplified by those languages which have maintained
the use of the article. The article, which had everywhere
originally the value of a demonstrative pronoun, marks the
word which it accompanies as the name of something to
which we point ; and what we point to is something which
admits of being observed by another person as well as by
ourselves. This can be done most easily with things which
have an actual position in space between the speakers ; but
developed language makes an object of any other matter
of thought in the same way. Such objectivity, therefore
(which in these cases also is indicated by the article), does
not entirely coincide with the reality which belongs to
things as such ; it is only the fact of their claiming such
a reality, on the ground of the distinctive peculiarity of
their real nature, which language has met and expressed
in their names. When we speak of 1 'the tooth-ache/
'the day,' 'the franchise,' we do not imply that they could
exist if there were no person to feel, to see, to enjoy them,
respectively. Still less when we talk of 'the adverb' or
'the conjunction,' do we mean to indicate by the article
that the subject-matter described by these words has any
sort of existence outside thought. We only mean that
certain special forms of resistance and tension, which we
feel in the course of our ideas, are not only peculiarities of
our own state and inseparable from it, but that they
depend upon relations inherent in the matter of various
ideas, which every one who thinks those ideas will find in
them just as we do.

The logical objectification, then, which the creation of
a name implies, does not give an external reality to the

1 [The instances in the text are der Schmerz, die Hclligkeit, die
Freiheit, but none of the equivalents are used in the required sense with
the article in English. The same applies to the instances in the follow-
ing sentence, das Zwar, das Aber, das Dennock.']

1 6 THE THEORY OF THE CONCEPT. [Book I.

matter named; the common world, in which others are
expected to recognise what we point to, is, speaking
gene\.;]ly, only the world of thought ; what we *do here is
to ascribe to it the first trace of an existence of its own and
an inward order which is the same for all thinking beings
and independent of them : it is quite indifferent whether
certain parts of this world of thought indicate something
which has besides an independent reality outside the
thinking minds, or whether all that it contains exists only
in the thoughts of those who think it, but with equal validity
for them all.

4. But the objectification of the matter so first constituted
is not the whole of this first act of thought ; consciousness
cannot simply present the matter to itself, it can only do so
by giving it a definite position ; it cannot simply distinguish
it from an emotional mood of its own, without accrediting
it with some other sort of existence instead of that which
belonged to it as such a mood. The meaning of this
requirement (for I admit that my expression of it is not
immediately clear) is most simply shown by the way in
which language actually satisfies it. It is only in the
interjection, which is not a name of definite content, that
language retains the formlessness which belongs to it as the
mere expression of excitement ; the rest of its stock of
words is articulated in the definite forms of substantives,
adjectives, verbs, and the familiar parts of speech in general.
And it is hardly necessary to insist that the various char-
acters thus impressed by language upon its material are the
indispensable condition of the later operations of thought ;
it is obvious that neither the combination of marks into
the concept, of concepts into the judgment, or of
judgments into the syllogism would be possible, if the
matter of every idea were equally formless or appre-
hended in the same form, if some of them were not
substantival and did not express fixed and independent
points of attachment for others which are adjectival, or if

Chap. I.] SUBSTANTIVES, VERBS, ADJECTIVES. 17

others again were not verbal, exhibiting the fluid relations
which serve to bring one thing into connexion with another.
I do not ' think it advisable to separate this pabular
conformation of the matter of ideas, as a second act of
thought, from the first act, to which we ascribed its ob-
jectification ; I prefer to comprise the primary activity of
thought in a single operation, which may be indifferently
represented as that of giving to the matter of ideas one
of these logical forms by making it objective for conscious-
ness, or as that of making it objective by giving it one
of these^forms.

5. The three parts of speech which I have noticed
remind us inevitably of three concepts which are indis-
pensable for our judgment of reality. It is impossible to
have even an expressible idea of the world of perception,
without thinking of things in it as fixed points which serve
to support a number of dependent properties, and are
connected together by the changing play of events. If
metaphysic is the investigation, not of the thinkable in
general, but of the real or that which is to be recognised
as such, these concepts of thing, property, and event are
metaphysical concepts; not perhaps such as metaphysic
would finally allow to stand without modification, but cer-
tainly such as at its outset purport to represent immediately
the proper essence and articulation of what is.

It would seem at first sight that the logical forms of
substantivity, adjectivity, and verbality coincide with these
concepts : but a second view shows the same difference
between the two series as that which separated the logical
objectification of an idea from external reality. Nothing
passes with us for a thing or a substance which has not
reality outside us and permanence in time, producing
changes in something else and capable of undergoing
changes itself; but we apprehend as substantives not only
things but their properties; as substantives we speak of
* change/ * occurrence,' even of * nothing/ and so in in-

T,OGIC, VOL. I. C

1 8 THE THEORY OF THE CONCEPT. [Book I.

numerable cases of that which has no existence at all or
no\^ except in dependence on something else. 9 Thus the
substantival form invests its content, relatively to the future
predicates to which it is to serve as subject, with only
the same priority and independence as belong to a thing
in contrast with its properties, conditions, and effects, but
by no means with that concrete and independent reality
and activity which place a thing above a mere object of
thought.

Verbs, again, express most frequently an event which
as a fact takes place in time ; but when we say that things
'are/ or 'are at rest,' or that one ' conditions' or 'equals 7
another, it is clear that the verbal form too does not
universally give to its content the meaning of an event,
but only finds it there usually. In order to conceive fully
the sense of such verbs as we have just instanced, we
have to connect several distinct contents together by a
movement of thought, and this movement though it implies
time for its execution, is, as regards its meaning and in-
tention, quite independent of time. In a word, the general
sense of the verbal form is not an event, but a relation
between several related points ; and this relation may just
as well occur between contents which are out of time and
coexist only in thought, as between those which belong
to reality and are accessible to temporal change.

Lastly, while it is true that radical adjectives, such as
'blue' and 'sweet,' express primarily what appears to our
first apprehension as a real property of things, every de-
veloped language knows words like 'doubtful/ 'parallel,'
'allowable,' which, as the least reflexion shows, can no
longer mean in the same simple sense as the former a
property attaching to actual things; they are abbreviated
and condensed expressions of the result of all sorts of
relations, and it is only for purposes of thought that we
represent the contents of such adjectives as related to
those of substantives in the way in which we imagine an

Chap, I.] THOUGHT AND SPEECH. 19

attribute to be related to its subject. Speaking generally,
then, the Jogical import of the parts of speech is o**.iy a
shadow of the import of these metaphysical concepts :
it only repeats the formal characteristics which the latter
assert of the real; but by not confining their application
to the concrete external reality, it loses that part of their
significance which they only possess in that application.

6. Lastly, if we found in the forms of the parts of speech
the most original activity of thought, we must also under-
stand how to distinguish this from its linguistic expression.
Now that man has come to use the language of sounds
for the communication of his thoughts, that activity is, it
is true, most clearly manifested in the forms of the parts
of speech ; but in itself it is not inseparably bound up
with the existence of language. The development of which
the ideas of the deaf and dumb are capable, though guided
in the first instance by those who can speak, is enough
to show that the internal work of logic is independent of
the possibility of linguistic expression. That work consists
merely in the fact that we accompany the content of one
idea with the thought of its comparative independence,
while we think of another as requiring support, and of
a third as a connecting link which neither subsists on
its own account nor rests upon something else but me-
diates between two others. No one doubts the extremely
effective support which language gives to the development
of thought by making the formations and transformations
of ideas vividly objective to consciousness by means of
sharply defined sounds and their regular changes; still, if
some other mode of communication were natural to man
instead of the language of sounds, the same logical asso-
ciations would find in it a corresponding expression though
of a different kind. And if in some languages the poverty
of forms does not always allow these associations to take
shape, cannot, for instance, distinguish between substantival
and verbal construction, yet there is no doubt that the mind

C 2

20 THE THEORY OF THE CONCEPT. [Book I.

of those who speak them maintains the logical distinctions
wh.^e forming ideas which are vocally undistinguished.
Whertver there is this inward articulation, there is thought;
where it is wanting, there is no thought. For this reason
music is not thinking ; for however manifold and delicately
gradated are the relations of its tones, it never brings them
into the position of substantive to verb, or into dependence
such as that of an adjective on its noun or a genitive on
the nominative by which it is governed.

7. In mentioning hitherto only three out of a greater
number of parts of speech, the three without which the
simplest logical enunciation would be impossible, I do not
wish to deny the value of the others. But the road which
we have to traverse is too long to allow us to make further
circuits into the attractive field of philological enquiry,
circuits which, considering how thought has just been said
to be independent of its mode of expression, must for our
purpose remain circuits. The articulation and usage of
language do not fully cover the work of thought. We
shall find later that they frequently do not express the
complete structure of the thought ; and then we have for
the purposes of logic to supplement what is said by what
was meant. On the other hand language possesses technical
elements which do not depend, or only depend with various
degrees of remoteness, upon characteristics essential to
logic : in such cases we should not be justified in distin-
guishing a different logical operation of thought for every
grammatical or syntactical difference of form presented by
language. There are not only interjections, but particles
too, which, like the tone of the voice, hardly indicate more
in ordinary usage than the interest which the speaker feels
in what he is saying, and contribute nothing to its sub-
stantial logical meaning. When language introduces the
distinction of gender into all substantives and adjectives,
it follows an aesthetic fancy which has no interest for logic ;
when on the other hand it determines the gender of the

Chap. I.] REMAINING PARTS OF SPEECH. 21

adjective by that of its substantive, this consistency in an
arbitrarily adopted custom points back to a logical relation-
ship which we shall become acquainted with. When : A the
inflexions of the verb it distinguishes the person speaking
from the person spoken to and the third person not present,
it emphasises an extremely important fact in a way which
is indispensable for the living use of speech, and yet there
is no corresponding distinction in logic proper. It is
nothing but the same reason which justifies grammar in
considering pronouns as a specific class of parts of speech :
logically;, the personal pronouns must be reckoned entirely
among substantives, with which in formal position they are
identical; the possessive and demonstrative we have no
ground for separating from adjectives; the relative we
should regard as the most specifically technical element in
language, serving only the need of methodical communica-
tion, and based on no other logical relation than its counter-
part the demonstrative. Numerals are treated by grammar
as distinct parts of speech ; in the actual usage of language
they are equivalent to adjectives, and that logically they
belong to the latter we cannot doubt, when we remember
that logically the form of adjectivity belongs to all charac-
teristics of a subject-matter which are not self-dependent,
and not only to those which attach to it in the sense of
properties. Adverbs, lastly, stand in precisely the same
relation to the meaning of verbs as adjectives to that of
substantives, so that logic would have no occasion to con-
sider them as a distinct part of speech or a peculiar form
of the content of thought.

Thus there would only remain prepositions and con-
junctions to put forward such a claim, and of them I think
we must admit that, however they may be derived linguisti-
cally, they form an indispensable element in the world of
our ideas. They cannot be derived from the concept of
relation, with which at first they seem to be connected :
whenever two members are connected by a relation, there

22 THE THEORY OF THE CONCEPT. [Book I.

is involved the thought of a certain position which those
members occupy within the relation itself, and this position
neea not be the same for both; on the confrary, it is
generally different, the one embracing, containing, and
conditioning the other. Now it will be found upon trial
to be impossible to express this difference of value between
the related points, without which the relation has no
meaning, in a merely verbal form : somewhere or other
we shall need a preposition, a conjunction, or at least
one of the various case-forms in which many languages
express some of these accessory notions still more shortly.
In what linguistic form they appear, is of course quite
indifferent to logic ; just as we oppose the nominative, as
that which conditions, sometimes to the genitive, as that
which is conditioned, sometimes in a different sense to the
accusative, so, if language had produced or preserved a
still greater wealth of cases, all prepositions would be
superfluous, as all conjunctions would be if there were a
similar variety of moods. This would make no change in
the logical needs of thought; in one way or another, the
meanings of substantives, adjectives, and verbs would have
to be supplemented by a number of ideas, indicating, either,
like prepositions, the position of two supposedly simple
objects in a simple relation, or, like conjunctions, the com-
parative position and value of two relations or judgments.

8. If we glance at the developed structure of the world
of our thoughts and ask what the conditions are upon
which its construction depends, the objectification of im-
pressions and their concomitant formation in the sense
of the parts of speech must always appear as the most
indispensable, and in that sense the first, of all operations
of thought. It is certain that without it the framing of
sentences, simple or complex, through which we express
the work and results of our thinking, would have been
quite impossible. But we must not be taken to mean
that the logical spirit, at the beginning of its intellectual

Chap. I.] JUDGMENT PRIOR TO CONCEPT? 23

work, before it ventured a step further, performed this,
the first of its necessary operations, on the entire matter
of its ideus once for all. The infinitude of possible Im-
pressions, of which every moment may bring a new one,
would be enough to make such a task impracticable : it
is made still more impracticable by the fact that in work-
ing up the matter that is given to it thought is constantly
producing new matter, and has to bring this again into the
same logical forms of which, as applied to a simpler matter,
it is the result. Thus it is that every developed language
possesses, in the form of simple substantives, adjectives,
or verBs, numerous ideas which could not have been
framed, and cannot be fully understood, without manifold
intellectual operations of a higher kind, without employing
judgments and syllogisms, and even without presupposing
systematic scientific investigation.

This obvious reflexion has given rise to the assertion,
that in logic the theory of judgment at least must pre-
cede the treatment of concepts, with which it is only
an old tradition to begin the subject. I consider this to
be an over-hasty assertion, due partly to a confusion of the
end of pure with that of applied logic, partly to a general
misconception of the difference between thought and the
mere current of ideas. For if those judgments, out of
which the concept is said to result, are to be really judg-
ments, they themselves can consist of nothing but com-
binations of ideas which are no longer mere impressions ;
every such idea must have undergone at least the simple
formation mentioned above \ the greater part of them,
as experiment would show, will already practically possess
that higher logical form to which the very theory in question
gives the name of concept. The element of truth in this
proposed innovation reduces itself to the very simple
thought, that in order to frame complex and manifold con-
cepts, more especially in order to fix the limits within
which it is worth while and justifiable to treat them as

24 THE THEORY OF THE CONCEPT. [Book I.

wholes and distinguish them from others, a great deal of
preparatory intellectual work is necessary ; but that this
prep*, "itory work itself may be possible, it must have
been preceded by the conformation of simpler concepts
out of which its own subsidiary judgments are framed.
Without doubt, then, pure logic must place the form of
the concept before that of the judgment : it remains for
applied logic to tell us how, in framing determinate con-
cepts, judgments consisting of simpler concepts may be
turned to account. A proposal to reverse this order can
only commend itself to those who regard thinking in
general as merely the interaction of impressions excited
in us from without, and overlook the reacting energy
which makes itself felt at every point in the current of
ideas, separating the merely coincident, combining the
coherent, and thus already giving form to the individual
elements of future thoughts.

B. Position l , Distinction, and Comparison of the Matter
of Simple Ideas.

9. If we recognise in these first formative acts the specific
contribution which the operative energy of thought makes
to the whole of our intellectual world, we are easily led to
the view that the logical spirit has certain ready-made
modes of apprehension with which it meets the impressions
as they come ; and this again raises the question, how it
contrives to bring the matter of each impression under that
particular form which is appropriate to it. But such a view
is inadmissible, and such a question therefore has no point,
or at any rate leads to an answer different from that
which it expects. Thought does not stand fronting the
impressions as they arrive with a bundle of logical forms
in its hand, uncertain which form can be fitted to which
impression, and therefore needing some special expedient

1 [' Position,' as the equivalent of Setzung, is here used in the active
sense in which it occurs, e. g. in ' composition.']

Chap. I.] POSITION AND DISTINCTION. 25

to discover how to pair them properly. It is the relations
themselves, already subsisting between impressions when
we become conscious of them, by which the act 1 '- ii of
thought, which is never anything but reaction, is attracted ;
and this action consists merely in interpreting relations,
which we find existing between our passive impressions,
into aspects of the matter of the impressions. It is not
therefore the assignment of the proper form of each matter,
which requires any special device of thought : in another
point of view, however, this arrangement of a manifold
matter in logical forms does involve a second intellectual
operation; for no matter can have a name made for it
unless it has been thought of as identical with itself, as
different from others, and as comparable with others.

10. This second operation of thought, like the first, is
one which inherited language has already carried out for all
those who speak it ; like the first therefore it is easily over-
looked, and not reckoned as part of the work of the mind.
But logical science, expressly devoted to the self-evident,
must not treat a part of its subject as a still more self-
evident presupposition which may be excluded from the
proper objects of its consideration. Still, the first at any
rate of the three heads under which we expressed this new
operation of thought does not need a detailed explanation.
It is at once obvious how every name, ' sweet ' or t warm,'
* air ' or 4 light/ ' tremble ' or ' shine,' gathers up the matter
which it indicates in some sort of coherent unity with a
meaning of its own; it is not only (though it is most
emphatically) matter in the substantival form that is thus
lifted into unity with itself by the prefixed article; the
same indicative force resides, under a different form, in the
infinitive of the verb, and even when language has no
distinctive expression for it, this accessory notion of
singling out and giving position to the matter indicated
accompanies every form of word. It may be doubted
whether the process which we would understand by giving

26 THE THEORY OF THE CONCEPT. [Book I.

position is not already contained in the objectification which
we represented as converting the passive impression into
an\"^ea; and it is true that we can neither have an idea
without thus giving position to its content, nor give it
position in any intelligible sense without objectifying it.
Practically therefore it is a really inseparable operation
which we are considering from different sides ; before, we
contrasted the presented idea to which we are related as
presenting, with the impression by which we are simply
affected; now, when the multiplicity of the matter pre-
sented begins to excite our attention, we lay stress upon
the unity and independence in virtue of which the matter
thus singled out by attention is what it is and differs from
everything else.

11. By the last words I wished to convey clearly the
close connexion in which the affirmative position given to a
content stands with the negative exclusion of all others.
The connexion is so close, that the terms which we are
obliged to employ to express the simple sense of the first
are only made perfectly clear by adding the accessory notion
of the second. We can only explain what we mean by the
unity of position given to a content by emphasising its
difference from others, and saying, not only, it is what it is,
but also, it is not what others are. The affirmation and the
negation are one inseparable thought, and accompany in
inseparable union every one of our ideas, even when we do
not expressly attend to the others which are tacitly negated.
But the accessory notion thus amalgamated with our ideas
only determines the logical setting which we give to their
content; it does not produce that content in the first in-
stance. It cannot be said that we have the idea of red as
red only when we distinguish it from blue or sweet, and
only by so distinguishing it, and again the idea of blue as
blue only by a similar opposition to red. There could be
no conceivable occasion for attempting such a distinction,
nor any possibility of succeeding in the attempt, unless

Chap. I.] COMPARISON. 2?

there were first a clear consciousness of what each of the
two opposites is in itself. Without doubt the peculiar
impression which we experience under the influe^e of
red light will be entirely the same before we have had our
first experience of blue light as it will be afterwards; the
possibility of comparison and distinction which the latter
experience gives may indeed, at any rate in a matter more
complex than these simple colours, draw the attention to
parts of the impressions which had been previously over-
looked, and so make both of them more complete; but
even in this case, which is quite outside our present con-
sideration, the new element is not discovered by the dis-
tinction, but by the immediate sensation of which the
comparison was merely the occasion. It is always affirma-
tive position therefore which makes negative distinction
possible, while it is never the case that the act of distinction
gives rise to the matter distinguished. Only our accessory
notions about the matter of our ideas, only its logical
setting, gains in definiteness by adding to the affirmation
of itself the negation of others ; and even this gain would
seem to me small if it went no further, and were not sup-
plemented by that third operation of positive comparison,
which, in the above account of this second act of thought,
was mentioned last. ,

12. I will introduce the consideration of this third opera-
tion, which I regard as the most essential part of the logical
work to be here explained, by recalling a familiar fact
which is commonly used to support other conclusions.
Words never denote impressions as they can be experienced ;
we can only experience or actually perceive a particular
shade of red, a specific kind of sweetness, a definite degree
of warmth, not the universal red, sweet, and warm of
language. The universalisation which in these and all
similar cases the matter of sensation has undergone, is
commonly regarded as an unavoidable inexactness of
language, perhaps even of the thought which language

28 THE THEORY OF THE CONCEPT, [Book I.

serves to express. Unable or not accustomed to make a
definite name for every single impression, language (it is
supposed) blurs the slight differences between them, and
retains only what is immediately experienced in sensation
as common to them all : by this reduction of its means of
expression to a moderate number it certainly makes the
communication of ideas possible, but diminishes propor-
tionately the exactness of that which has to be communi-
cated. I do not think that this view does full justice to
the significance of the fact.

13. First of all, to regard the universalisation in question
as a sort of falsification of the impressions is to pass too
lightly over the very remarkable circumstance, that in a
number of different impressions there is something common
which can be thought apart from their differences. This is
by no means such a matter of course that the opposite is
out of the question ; on the contrary, it is quite conceivable
that every one of our impressions should be as incomparably
different from every other as sweet actually is from warm,
yellow from soft. The fact that the thinkable world itself
is so constituted that this is not the case, is one which it is
worth while to take into consideration. Nor again can I
regard the want of exactness, which the application of the
universal terms of language undoubtedly gives to the com-
munication of ideas, as sheer loss. Moreover, when perfect
exactitude is felt to be important, the shortcomings of these
simplest products of rudimentary thought can always be
supplemented by its more advanced activity : science has
long taught us to measure every degree of heat, and in case
of necessity would find out how to measure every gradation
of redness or sweetness.

But the way in which language and natural thought
operative in language solve the same problem, seems to me
to be logically very significant. For when, instead of
attaching a particular name to every single colour of which
we have actual sensation, we give the privilege of names of

Chap. I.] THE ' FIRST UNIVERSAL. 29

their own to blue, red, yellow, and a few others, and then
intercalate the other individual sensations between them as
bluish red or reddish yellow, this is not merely a shlit for
approximating to an unattainable exactitude ; rather, as it
seems to me, it expresses the conviction that only these few
colours are really fixed points deserving names of their own,
while the rest must be characterised by approximate ex-
pressions because they are themselves only approximations
to these fixed points, or connecting links between them.
If we really had particular and mutually independent names
for every single shade of blue, and our ideas answered to
this form of expression, we should have achieved in a one-
sided way the separation of each from every other, but we
should have overlooked completely the positive relations
which subsist between them all. If on the contrary we
speak of bright blue, dark blue, black blue, we arrange this
manifold in a series or a network of series, and in each series
a third member results from a second by intensification of
the same sensible change in a common element as that
which gave rise to the second out of the first. It must be
already perfectly clear that a presentative activity which did
not involve this comparison of the diverse, but was confined
to the bare separation of each from each, would not offer to
the later operations of thought adequate grounds for con-
trasting two ideas, as in some way or other cohering, with
two others as not cohering. We therefore apprehend this
second act of thought, of which we are here speaking, not
merely as that of giving simple position to a or , not merely
as that of simply distinguishing every a from every , but
also as that of determining the extent and peculiarity of the
distinction, which is not everywhere the same in degree and
kind, but is different between b and c and between a and b.
I do not mean to say that every single idea, a, must be
accompanied by the developed idea of all its relations to
the infinite number of all other ideas ; the general accessory
notion, that every idea is enclosed on all sides in such a

30 THE THEORY OF THE CONCEPT. [Book I.

network of relations, does indeed in our logical conscious-
nes envelop every idea ; but these relation^ are only
followed out in each particular case so far as a special
requirement suggests.

14. This comparison of the diverse clearly presupposes a
common element to which in the several members of the
series specific differences attach. Such a common element
is usually considered by logic only in the form of a uni-
versal concept, and in this shape it is a product of more or
less numerous acts of thought. It is therefore important to
point out that this first universal, which we find here in-
volved in the comparison of simple ideas, is of an essen-
tially different kind ; that it is the expression of an inward
experience which thought has merely to recognise, and that
just for this reason it is, as will be seen later, an indispens-
able presupposition of that other kind of universal which
we shall meet with in the formation of concepts. We im-
part the universal concept of an animal or a geometrical
figure to another person by directing him to execute a
precisely definable series of intellectual operations, con-
necting, separating, or relating a number of simple ideas
assumed to be known ; when this logical work is completed,
we suppose him to have before his mind the same object-
matter which we wished to impart to him. But we cannot
explain by the same means wherein the universal blue or
the universal colour consists, which accompany our ideas of
bright and dark blue or of red and yellow. We can indeed
direct another person to think of all single colours or all
shades of blue, and by eliminating their differences bring
out what is common to his ideas in the two cases ; but it is
only in appearance a logical work which we are here pre-
scribing ; all that we really call upon him to do is to see for
himself how he executes the task. How he is to set to
work to discover whether there really is any common
element in red and yellow, and how he is to contrive to
separate it from the differences, this we cannot tell him ; we

Chap. 1,1 THE UNIVERSAL NOT AN l IDEA? 31

must simply trust to his having an immediate sensation,
feeling, or experience of the connexion which exists bet^ieen
red and yellow, of the fact that they contain a common
element ; his logical work can consist only in the recognition
and expression of this inward experience. This first uni-
versal, therefore, is no product of thought, but something
which thought finds already in existence.

15. I will insert an observation here which with slight
modification may be extended to all universals, but is most
easily illustrated in this simplest instance, the first universal.
That in which red and yellow agree and which makes them
both colours cannot be separated from that which makes
red red and yellow yellow, not separated, that is to say, so
as to form the content of a third idea similar in kind and
order to the two compared. It is always, as we know, only
a single definite shade of colour, only a tone of definite
height, strength, and quality, which is the object of sensa-
tion ; and it is only these definite impressions which are so
repeated in memory as to present substantial and perceptible
images to consciousness. Universal ideas never have this
perceptibility. If we try to apprehend the universal element
of colour or tone, we shall always find that either we have
before our perception a definite colour and a definite tone,
only with the accessory notion that every other tone and
colour has an equal right to serve as a perceptible instance
of the ever imperceptible universal; or else our memory
will produce a number of colours and tones in succession,
with the same accessory notion that it is not these individuals
that are really meant, but the common element in them
which cannot as such be apprehended in perception. If
therefore we understand by idea 1 (as ordinary usage cer-
tainly inclines us) the consciousness of something standing
at rest before the mind, or a perception of something
capable of being presented to it, the universal cannot claim
to be called an idea. Words like ' colour ' and ' tone ' are
1 [Vorstellung.]

32 THE THEORY OF THE CONCEPT. [Book I.

in truth only short expressions of logical problems, whose
solution cannot be compressed into the form pf an idea.
They are injunctions to our consciousness to present to
itself and compare the ideas of individual tones and colours,
but in the act of so comparing them to grasp the common
element which our sensation testifies them to contain, but
which cannot by any effort of thought be really detached
from their differences and made the material of a new and
equally perceptible idea.

16. Let us now direct our attention to the differences,
which, within the first universal, separate the various
instances of it. It is clear that what distinguishes one
sensation of warmth from another, a gentler from a louder
sound, bright from dark blue, is a more or a less of a
common sensible element, which in itself, undetermined by
any degree, is no object of perception. We shall find
ourselves brought back to the same ground of distinction
in all other ideas ; it is only in giving an account of the
universal, to which this quantitative comparison applies,
that we meet with a difficulty, which after the above
remarks is intelligible. The louder tone is no doubt
distinguished from the gentler by a certain intensification,
but so also is the higher from the lower ; yet it is only in
the former case that we feel able to express directly, by the
term * strength/ the common element which undergoes Jhis
change; in the latter we express it by the metaphor of
height. Red and yellow seem to be still more essentially
different and underivable one from the other by increase or
decrease of a common element; only the intermediate
colours, reddish yellow or yellowish red, are intelligible to
us as mixtures containing more or less of one or the other.
Nevertheless no one denies that one of the fundamental
colours is more nearly related to a second than to a third,
red to yellow than to green ; and these grades of resem-
blance cannot be conceived without a more or a less of some
common element, which we are conscious of in passing

Chap. I.] OBJECTS OF THOUGHT NOT FIXED, 33

from one member of the series to the next and from this to
the third. To determine in each particular case what this
common element consists in, to decide whether a nurr&er
of ideas are separated merely by differences in degree of
one simple universal, or by differences in value of several
mutually determined ones, and whether accordingly the
ideas are to be grouped in a linear series or plane-wise or in
still higher forms, these are all attractive objects of enquiry,
but they are not objects of logic. For logic it is enough to
know that some generally applicable and primarily quantita-
tive determination is the indispensable means for dis-
tinguishing between the particular instances of a universal.
And even this determination is something which it is not
the work of logic to produce, but only to find, recognise,
and develop. A judgment, a is stronger than bj is indeed,
as a judgment, a logical piece of work ; but that which
it expresses, the general fact that differences of degree
do exist in the same matter, as well as the particular fact
that the degree of a exceeds that of <, can only be ex-
perienced, felt, or recognised as part of our inward con-
sciousness. By whatever artificial contrivances we may
seek to increase scientifically the exactness of a measurement,
everything must depend ultimately on the capacity to recog-
nise two sensuous perceptions as like or as unlike, and not to
be deceived as to which has the more and which the less.

17. If inward experience were confined to bringing out
resemblances and differences in the various object-matters,
thought would merely be called upon to arrange ideas in an
unalterable system, like the musical scale, in which all tones
have once for all their fixed and immoveable places. But
logic has to do with thought, not as it would be under hypo-
thetical conditions, but as it is. Now owing to the mechan-
ism which controls the interaction of its inward states, all
actual thought has necessarily more opportunities of stimula-
tion than the above hypothesis would imply ; the manifold
matter of ideas is brought before us, not only in the

LOGIC, VOL. I. D

34 THE THEORY OF THE CONCEPT. [Book I.

systematic order of its qualitative relationships, but in the
rich variety of local and temporal combinations ; and this
fac^ like the other, belongs to the material wnich serves
thought in its further operations and must be given it to
start with. The combinations of heterogeneous ideas pro-
duced in this way form the problems, in connexion with
which the efforts of thought to reduce coexistence to co-
herence will subsequently have to be made. The homo-
geneous or similar ideas on the other hand give occasion
to separate, to connect, and to count their repetitions ; and
to these ideas of unity and multiplicity those of greatness
and smallness are added where the matter presented is
continuously extended in space or in time. These three
pairs of quantitative ideas (for we have already got those of
more and less) comprise all the standards by which the
individual instances of any universal are distinguished.

18. There are two things which I intentionally exclude
from my consideration. Firstly, all enquiry into the psycho-
logical character of the growth and development of these
quantitative ideas in our consciousness, into the order in
which one of them may condition the origin of another, and
into the different importance of perceptions of time and
space in their formation. However attractive these ques-
tions may be, it would lengthen our way unnecessarily
to answer them ; logic is not concerned with the manner in
which the elements utilised by thought come into existence,
but with their value, when they have somehow or other
come into existence, for the carrying out of intellectual
operations. Now this point, which I conceive to have been
unduly neglected, I wish to emphasize here, and shall
subsequently keep in view, viz. that all ideas which are to
be connected by thought must necessarily be accessible
to one, of the three quantitative determinations which have
just been mentioned. The other thing which I exclude
is the investigation of the consequences which may be
drawn from these quantitative determinations as such : they

Chap. I.] SPONTANEITY AND RECEPTIVITY. 35

have long ago developed into the vast structure of mathe-
matics, the complexity of which forbids any attempt to
re-insert it in universal logic. It is necessary, however,
to point out expressly that all calculation is a kind of
thought, that the fundamental concepts and principles
of mathematics have their systematic place in logic, and
that we must retain the right at a later period, when
occasion requires, to return without scruple upon the results
which mathematics have been achieving, as an indepen-
dently progressive branch of universal logic.

10. If we take a general survey of this second act of
thought, in which I now include that of giving affirmative
position to the object-matter, that of distinguishing it nega-
tively from all others, and that of estimating by quantitative
comparison its differences and resemblances, we may ob-
serve that the significance of this new logical operation
is somewhat different from that of the first, by which
impressions were shaped into ideas. In the former case
there was a temptation (which, it is true, we resisted)
to regard the forms of substantivity, adjectivity, and ver-
bality as modes of apprehension which thought is ready
to put in practice upon its object-matter before receiving
any solicitation from it ; but though we set aside this claim
at once, it remains true that in those forms thought does
not merely respond to and reproduce the actual current
of ideas, but gives them the shape without which the logical
spirit could not accept them. The independence which
the substantival form gives to its matter, most obviously by
means of the article, did not itself lie in the fact that
this matter was a permanent link between changing groups
of ideas ; nor was the accessory notion of dependence
expressed by the adjectival form present, as such, in the
fact which stimulated the mind to characterise it by that
form ; so that we may continue to assert, in a certain sense,
that in this first act thought dictates its own laws to its
object-matter.

D 2

36 . THE THEORY OF THE CONCEPT. [Book I,

If, using an expression which we shall otherwise avoid,
we represent this procedure as a proof of spontaneity, the
second act of thought has the character of receptivity; it is
a recognition of facts, and adds no other form to them
except this recognition of their existence. Thought can
make no difference where it finds none already in the matter
of the impressions ; the first universal, as we saw, can only
be experienced in immediate sensation ; as so experienced
it can be named, but this is the only contribution which
logic can make to the further fixing of its character; all
quantitative determinations, to whatever extent thought
may develop them by subsequent comparison, always come
back to an immediate consciousness of certain characteris-
tics given in the object-matter. I should wish this fact to
be considered from two points of view. In the first place,
logic is guilty of a certain carelessness in assuming at almost
every moment in its later stages the comparability of ideas
and the possibility of their subordination to a universal,
without observing that that possibility, and the success of
its own procedure in general, depends upon this original
constitution and organisation of the whole world of ideas, a
constitution which, though not necessary in thought, is all
the more necessary to make thinking possible. For I must
repeat that there is no inherent contradiction in supposing
that every idea was incomparably different from every other;
that in the absence of all qualitative comparability there was
no standard of more or less; that the same idea never pre-
sented itself twice to perception ; and that, as there was no
repetition of the homogeneous, the ideas of larger and
smaller also vanished. The fact that this is not the case,
but that the world of ideas is organised as we have found it
to be, must be emphasized as of the highest importance ;
but logic ought not in case of need to appeal to it inci-
dentally as a self-evident truth derived no one knows
whence. And this brings me to the other observation
which I had to make. If thought is a reaction upon a

Chap. I.] FORMATION OF CONCEPT. 37

stimulus found in the current of ideas, a systematic survey
of its functions will show clearly at certain points the in-
fluence exercised upon them by the thinkable world ; as it
is here the second member in the first triple series of opera-
tions, so at a later stage also it will be the second member
of the following more highly developed group in which we
shall see the peculiar dependence of thought upon the
material to which it is directed. I do not however claim
to do more by this preliminary indication than to throw a
preliminary light over the system which I have followed in
my exposition ; the system itself can only find its justifica-
tion in the advantages which in its successive stages it will
be found to secure.

C. The Formation of the Concept.

20. To separate the merely coincident amongst the
various ideas which are given to us, and to combine the
coherent afresh by the accessory notion of a ground for
their coherence, is the further task of thought. It will be
useful, with a view to making its meaning clear, to rej/iew
the different senses in which any combination of manifold
elements occurs in our mental world. In the first place, no
later intellectual activity is possible, unless the various ideas
upon which it is to be exercised meet together in one and
the same consciousness. The fulfilment of this condition is
secured by the unity of the soul and the mechanism of
memory, which, by bringing together impressions separated
in time, makes their interaction possible. This union of
the manifold may be called the synthesis of apprehension ;
it is not a logical act; it merely lumps the manifold together
into a simultaneous possession of consciousness, without
combining any two of its elements in a different order from
any other two. Such an order comes in with the second
form of connexion, the synthesis of perception, that is, with
figures in space and succession in time, in which the in-
dividual impressions take up definite and non-equivalent

38 THE THEORY OF THE CONCEPT. [Book I.

positions. This connexion also is supplied by the inward
mechanism of consciousness without any action pf thought,
and however firmly defined and finely articulated it may be,
it exhibits nothing but the fact of an external order, and
reveals no ground of coherence justifying coexistence in that
order. From the second stage I pass at once to the fourth,
to a synthesis in which the last-mentioned requirement
would be completely satisfied in regard to any given object-
matter. In such a synthesis we should have before our
mind, not the mere fact of manifold elements in order, but
also the value which each element possessed in determining
the coalescence of the whole. If what we thus apprehended
were an object in real existence, we should see which were
the prior, determining, and effective elements in it, in what
order of dependence and development the others followed
from them, or what end was to be regarded as their autho-
ritative centre, involving in itself the simultaneous union or
successive growth of them all : if, like the figures of geometry,
it was something which had no reality out of our conscious-
ness and no growth or development in time, we should here
too attempt at any rate (though, as we shall see later, with
limited success) to arrange the elements of the whole in a
hierarchy in which those that conditioned others should
take precedence of those that were conditioned, according
to their stages of dependence. It is easy to see that a
synthesis of this sort would be neither more nor less than
the knowledge of the thing ; as the goal of all intellectual
effort, it lies as far above the province of logic as the first
and second modes of connexion lay beneath it ; it is in the
space between that we must place the third and logical form
of synthesis, the character of which has now to be examined.
21. When a person who has no special knowledge speaks
of * credit' or of 'banking,' we trace in these expressions
his conviction that a number of businesses and institutions
form a connected whole; but he would not be able to
say where the nerve of the connexion lies, or what limits

Chap. I.I THE IMPERFECT CONCEPT. 39

separate the whole from that which does not belong to it.
In this accessory notion, that the various elements are not
merely there in a sort of heap, but form a whole of parts
with self-imposed limits and a unity included by those
limits, the general impulse of thought leaves its mark upon
the given object in a formal way, without as yet attaining
material fulfilment. If we pass our mental world in review,
we are in this position as regards a very large part of its
contents ; indeed we shall be surprised to find that words
of great significance betray this imperfect apprehension
of their objects ; for the more complex, important, and
various any matter is, the more easily will persuasive im-
pressions derived from repeated observations awaken the
feeling of its individuality, completeness, and self-inclusive-
ness, without necessarily giving any real insight into its
structure. Such words as ' nature/ * life,' ' art/ ( knowledge,'
' animal,' and many others have no more significance than
this in ordinary usage; they merely express the opinion
that a certain quantity, usually not exactly definable, of
individual objects, attributes, or events, which attach to
one another, form somehow an inwardly connected whole,
which can neither have any part taken away without being
destroyed, nor admit any casual additions within the bounds
of its unity. But how little the nature of this connexion
is really known, appears from the failure of the attempt
to describe the limits which include what belongs to the
unity and exclude what does not. So long as the logical
work of holding the manifold together has not gone further
than this, I should hesitate to speak of * concepts,' though
I do not attach any value to the invention of a special
technical term for such imperfect apprehension. Suppose
we call it an imperfect or growing concept ; then we shall
not feel that we have' got a perfect or fully developed
concept, until the vague suggestion of some sort of whole
has grown into the pervading thought that there is a definite
ground for the co-existence of these particular attributes,

40 THE THEORY OF THE CONCEPT. [Book I.

in this particular combination and to the exclusion of
certain others, and that this ground is an adequate one.

22. The question now arises, how we get at this ground
and condition. If we merely continued to observe a com-
posite form a b c d in its isolation, we should never discover,
however long we looked, which of its parts only coexist,
which really cohere, and in what degree the existence of
one depends upon that of another. But if we compare
abed with other forms like it, that is, with such as we
are led from it to observe, not by any special logical effort,
but by the natural current of our ideas, and if we find
that in abed, a b c f, a b c g, etc., a similar group a b c
occurs with various dissimilar additions, we regard the
latter as loose and separable appendages of the permanent
stem a b c. Nor does the common group a b c contrast
with the rest merely as the centre to which as a matter
of fact they attach ; on the general assumption that we
have before us a whole of interdependent parts, this solid
kernel becomes the expression of the constant rule which
allows the accretion of the several accessory elements, and
determines the manner in which it takes place. If we wish
for practical purposes to ascertain in any creature, object,
or arrangement, what is the line which divides what is
inwardly coherent from casual accessions, we put the whole
in motion, in the belief that the influence of change will
show which parts hold firmly together while foreign ad-
mixtures fall away, and in what general and constant modes
those parts combine while changing their relative positions
in particular cases : in this sum of constant elements we
find the inner and essential cohesion of the whole, and
we expect it to determine the possibility and the manner
of variable accretions. The first of these methods, that
of bringing out the common element in different instances
when at rest, is the one which has been usually followed
by logic, and has led to the formation of the logical
universal ; I should give the preference to the other, that

Chap. I.] THE NATURE OF ABSTRACTION. 41

of determining the element which maintains itself in the
same instance under changed conditions; for it is only
the assumption that the group a b c, the common element
in several groups of ideas, will also be found thus to
maintain itself, which strictly justifies us in regarding these
coexisting elements as coherent, and as the ground for the
admissibility or inadmissibility of fresh elements.

23. Abstraction is the name given to the method by
which the universal is found, that method being, we are
told, to leave out what is different in the particular instances
compared and to add together that which they possess in
common. If we look at the actual procedure of thought,
we do not find this account confirmed. Gold, silver,
copper, and lead differ in colour, brilliancy, weight, and
density ; but their universal, which we call metal, is not
found upon comparison by simply leaving out these differ-
ences without compensation. Clearly it is no sufficient
definition of metal to say negatively, it is neither red nor
yellow nor white nor grey; the affirmation, that it has at
any rate some colour, is equally indispensable ; it has not
indeed this or that specific weight, this or that degree of
brilliancy, but the idea of it would either cease to have
any meaning at all, or would certainly not be the idea
of metal, if it contained no thought whatever of weight,
brilliancy, and hardness. Assuredly we do not get the
universal image of animal by comparison, if we leave out
of our minds entirely the facts of reproduction, self-move-
ment, and respiration, on the ground that some animals
produce their young alive, others lay eggs, others multiply
by division, that some again breathe through lungs, others
through gills, others through the skin, and that lastly many
move on legs, others fly, while some are incapable of any
locomotion. On the contrary, the most essential thing
of all, that which makes every animal an animal, is that
it has some mode or other of reproduction, of motion, and
of respiration. In all these cases, then, the universal is

42 THE THEORY OF THE CONCEPT. [Book I.

produced, not by simply leaving out the different marks
p l and / 2 , q 1 and ^ 2 , which occur in the individuals com-
pared, but by substituting for those left out the universal
marks P and <2, of which /* / 2 and q^ q 1 are particular
kinds. The simple process of leaving out only takes place
when one of two individuals compared actually possesses
no species of a mark P, of which some species is a neces-
sary mark of the other. Thus we suppose, whether rightly
or wrongly does not matter, that we cannot find in plants
any trace of sensation and self-movement, both of which
are essential to all animals; we do therefore form the
universal idea of organic being from a comparison of plant
and animal by leaving out these marks without compen-
sation. If we went thoroughly into the facts, we should
perhaps find occasion, not indeed in this instance but in
many similar ones, to continue to ascribe two marks jointly
to both the objects compared, but to assume them to be at
zero in the plant, while in the animal they always occur in
an appreciable quantity. To express the matter somewhat
differently, it may be asserted from the point of view of
logic that compensation by the corresponding universal for
omission of individual marks is the regular rule of ab-
straction, while the uncompensated omission applies to
exceptional cases, where we can find no logically common
mark, of which the presence and absence of some individual
mark might be held to constitute different species. So
formulated, our rule of abstraction covers these cases of
mere omission; on the other hand, a rule which made
omission its sole starting-point could find no way to bring
in compensation afterwards ; and the importance of com-
pensation in forming the universal will be confirmed at
every step in the later stages of logic.

24. After the considerations urged in the preceding
section, the necessity of which to what was to follow will
now be clear, the apparent circle involved in the injunction
to form universals by putting together universals, will not

Chap. I.] FIRST AND SECOND UNIVERSALS. 43

give serious offence. We have seen that the universal
marks P and Q which we require here, the ' first universal '
of the section referred to, come to us without logical effort
as simple facts of observation in our mental life ; and just
for this reason they can be applied in building up this
second universal, which we do produce by logical effort.
That the yellow of gold, the red of copper, and the white
of silver are only variations of a common element which
we proceed to call colour, this is a matter of immediate
sensation; but to a person who could not be made
sensible of it, it could never be explained by logic either
that these particular impressions are species of this uni-
versal, or what is meant by a universal as such and the
relation of its particular to it. It is just this point to
which I would again draw attention here, that the im-
mediate perception of a first universal and the application
of some kind of quantitative ideas is the condition of the
formation of the second universal in all cases, not only in
those like metal where there is no difficulty in regarding
the marks of colour, brilliancy, and hardness as stable pro-
perties of that which they describe, but also where, as in
the case of the animal powers of reproduction and motion,
they are merely short adjectival descriptions of conditions
which we cannot think completely but by means of mani-
fold relations between various related points. It is easy
to convince oneself by an analysis (which I only leave the
observant reader to make for himself because it threatens
to be a long one) that all differences between animals, even
in these respects, issue ultimately in quantitative deter-
minations, whether of the force with which some identical
or similar process takes place in them, or of the number of
related points between which it takes place, or of the
variations in form to which it is liable owing to variations
in the number of these related points, the intimacy of their
relations, and their relative positions in space and time,
these last, like the rest, being measurable variations. If

44 THE THEOR Y OF THE CONCEPT. [Book I.

we take away this quantitative gradation and comparability,
which extends, though of course in different ways, to every-
thing, whether simple properties, or their relations, or com-
binations of events simultaneous or successive, the formation
of a universal by comparison of different groups of ideas,
would, at least in the sense in which it has any value for
thought, be impossible.

25. I will now mention some traditional technical ex-
pressions. If we provisionally give the general name of
concept (notio, conceptus) to the composite idea which we
think as a connected whole, the sum of individual ideas or
marks (notae) a, b, <r, d, etc., through which a concept S is
fully thought and distinguished from all other concepts 2, is
called its ' content ' (mater la) ; while its * extent ' (ambitus,
sphaerd) is the number of individual concepts j 1 , J 2 , s"*, etc.,
in each of which the content of S, that is, the group of
marks #, , c, d, in some one or other of their possible
modifications, is contained. The colour, , weight, <,
elasticity, d, and the like, would together form the content
of metal, S, while copper, /, silver, j 2 , gold, s\ and the like,
taken together, form its extent. It is usual also to speak
of the individual marks #, , c, as ' coordinated ' in the
content of S, and of the individual species s\ s\ /, as ' co-
ordinated ' in the extent of S : the relation of the species
s\ s 2 , 5 3 , to the universal itself which forms their genrs, is
called * subordination, 7 while both the species and the
genus are said to be ' subsumed ' under each of the
universally expressed marks, which make up the content
of S, and consequently also of s\ j 2 , s*. Lastly, it is asserted
that the extent and content of every concept vary inversely ;
the greater the content, that is, the number of marks which
the concept imposes upon all its subordinate species, the
smaller is the number of species which fulfil this require-
ment; the smaller the content of -S 1 , the greater is the
quantity of individuals possessing the few marks necessary
to make them species of S or bring them within its extent.

Chap. I.] SINGULAR AND UNIVERSAL CONCEPT. 45

If therefore we compare the universal concept S with a
similar universal T> and look for a third universal U to
which both of them belong as species, and if we continue
this process, the higher each universal concept /^stands in
the scale, that is, the farther it is removed from the
concepts S and T originally compared, the poorer will it be
in content and the larger in extent ; and conversely, if we
descend from the highest universals /^through Fand 7,
S and T, to the species of S and lower, the content will
increase with the decreasing extent and become greatest in
those completely individual ideas to which logic hesitates
to give the name of concept at all.

26. The value of these distinctions is unequal, but on
the whole slight. I will begin what I have to say about
them by fixing the terminology which I shall myself use in
future. I speak of any composite matter s as conceived or
as a concept, when it is accompanied by the thought of a
universal S, which contains the condition and ground of
the coexistence of all its marks and of the form of their
connexion. After this explanation we shall not hesitate to
speak of concepts of perfectly individual things (singular
concepts, in the old logical terminology), and we believe
this to be quite consistent with the usage of language. For
when we observe a new object s for the first time, and, not
content with the perfectly clear sensible perception of it, go
on to ask what it really is, we clearly want to know the rule
which connects the perceived marks in the observed fact
and converts them into a coherent whole of a definite and
predictable character. If we then find that this s is S, an
animal or a plant, we suppose ourselves to have a con-
ception of s ; it is the idea of it which is raised into a
concept by the accompanying thought of the universal S.
Every proper name is an illustration of this. 'Alcibiades,'
for human thought, never means merely a multiplicity of
differently coloured points, which are combined in space in
a definite though not quite invariable outline, and resist the

46 THE THEORY OF THE CONCEPT. [Book I.

attempt to separate them ; nor does the name express
merely the accessory notion that this multiplicity in some
unexplained way forms a whole ; it suggests to the mind a
definite general image of a man or a human being, which
lays down the lines for our view of the connexion of the
observed marks with one another and with the future
behaviour to be expected from them. A view so deter-
mined cannot be appropriately called either a perception,
or an idea merely, but only a singular concept.

27. On the other hand, it seerns to me quite out of place
to call the universal S itself, the accompanying thought
of which makes the individual into a concept, without any
reservation a universal concept, S may have the form of a
concept, but by no means always has it ; often it remains a
mere general image, the thought of which is indeed accom-
panied by the thought of its connected wholeness, but does
not exhibit the organic rule of the connexion. The name
' man ' as ordinarily used expresses no more than an image
of this kind ; reflexion, by subordinating it to the universal
( animal,' easily makes it into a concept ; but then * animal '
remains a general image, which only the naturalist, for the
uses of his science, converts into a concept by thinking
{ organic being ' along with it. It is upon such incomplete
logical activity, which brings into relief only a single link in
the chain, the connexion of the individual with its nearest
universal, but leaves all beyond it in darkness, that the
concepts which occur in ordinary thinking are based; as
however scientific investigations, to which logic is primarily
intended as an introduction, do really aim at extending the
conceptual form from the concept itself to the higher
universals under which it successively falls, it is enough to
have made the above remark without rigidly enforcing it,
and I shall follow ordinary usage in conceding the name
of concept to those general images as well. I can do this
the more easily because the name ( concept ' does not seem
to deserve in logic that exalted significance which the

Chap. I.] COORDINA TION. 4 7

school of Hegel has given it, and in which it claims to
express the knowledge of the essential nature of the object.
The difference between logical forms and metaphysical ideas
must be taken into account here as elsewhere. There may
be a privileged concept, which follows the thing itself in its
being and development, or takes up a point of view at the
very centre of the thing, the fountain-head of its self-deter-
mination and self-organisation ; but it is not the function of
logic to reserve its concept-form for so very select a filling.
By the logical concept we understand such a form of
apprehending any matter of thought, from whatever point
of view, that consequences admit of being drawn from it
which coincide again at certain points with results flowing
from that matter, that is, from the thing itself ; and as the
thing projects itself differently at every different point of
view, there may be various equally right and equally fruitful
logical concepts of the same object. We may therefore
continue to call * concept ' any apprehension which, though
only with the help of a general image which is not further
analysed, has the effect of bringing the given object under
a rule of behaviour which agrees, when applied, with its
actual behaviour.

28. The asserted coordination of marks in the content
of the concept raises serious difficulties. To begin with,
it is a misfortune that we have no appropriate name for the
elements of which we compose the concept ; for ' mark J
and ' part ' only apply in certain cases. They give rise to
the current delusion that the elements of a concept are
universally of equal value, connected in the same way each
with the whole and each with each. The ordinary instances
of logic, taken from simple natural objects, are specially
calculated to lead us into this error. It is true that gold is
yellow only in the light, ductile only under a certain power
of traction, heavy only for the body upon which it presses ;
but these various modes of behaviour easily present them-
selves to our imagination as stable properties, collected in a

48 THE THEOR Y OF THE CONCEPT. [Book I.

definite point of space, and inhering, in a manner identical
but otherwise unexplainable, in the reality which on their
account we call gold. Here the name * marks' is appro-
priate, and here the marks are certainly coordinated in the
content as has been asserted ; but this coordination merely
means that they are all equally indispensable to the whole,
but have not any other sort of order. If we leave such
simple instances, and consider concepts like 'triangle,'
* animal,' or 'motion,' we require, in order to think them
properly, a quantity of part-ideas which are no longer
mutually equivalent, but have to be placed in the most
various relations to one another. The three sides of a
triangle are not merely there as well as the three angles ;
they must form the angles by their intersections : the concept
of motion does not merely contain the part-ideas of place,
change, direction, and speed ; direction and speed are, each
in a different sense, determinations of change ; place, being
that which is left behind, can least of all be called a mark
of the concept ; it is a point of reference for the idea of
change, to which its relation is expressed by that of the
genitive to the nominative which governs it. To follow
out these points in detail would take too long, but it would
evidently lead us to the conviction that, as a rule, the marks
of a concept are not coordinated as all of equal value, but
that they stand to each other in the most various relative
positions, offer to each other different points of attachment,
and so mutually determine each other; and that an appropri-
ate symbol for the structure of a concept is not the equation
Sz=a + & + c+d, etc., but such an expression as S=J?
(a, b, r, etc.), indicating merely that, in order to give the
value of S, a, , c, etc., must be combined in a manner
precisely definable in each particular case, but extremely
variable when taken generally. If in any particular

instance S= a[t> G sin d~\ + (e }V~h, this formula, how-

&
ever foolish it would be if it professed to mean anything

Chap. I.] SUBORDINATION AND SUBSUMPTION. 49

more, would give a better picture than the above inadequate
formula of addition of the different ways in which the several
marks a, b, t, etc. contribute to the construction of S as a
whole.

29. No objection need be made to the coordination of
s 1 , s 2 , /*, copper, gold, silver, within the sphere of S, metal ;
on the other hand, attention should be drawn to the great
difference of value between the subordination of the species
to the genus, and that of the universal along with its
species to the universal marks a, b, (ductile, coloured, etc.).
The nature of the universal, metal, completely dominates
the nature of its species, gold and copper, and no property
of the latter escapes its influence ; many things are yellow
or red, but the glistening red and yellow of copper and gold
belong to metal alone ; many things are ductile, but the
amount and other peculiarities of the ductility exhibited by
gold and copper are heard of only in metals ; and only
metallity explains their degree of specific gravity. Similarly
the universal animal determines every property and every
movement of its species; animals move, grow, and rest
differently from plants and lifeless things. If we symbolise
the universal metal by a circle , the smaller circle of gold,
s\ lies entirely within it, and by the side of this, separate
from it but also completely inside S, the circles j- 2 , j 3 , copper
and silver. Applying differently two names which are
generally used as equivalents, I describe the true subordina-
tion to a dominant universal as subordination to the genus,
while I call the subordination of gold to yellow or ductile
subsumption under the mark. These universal marks obvi-
ously do not rule and penetrate the whole nature of gold ;
each of them expresses only one side of it, which it shares
with other objects of an entirely different kind, from which,
so far as logic can see, no sort of inferences can be drawn
as to the other properties of gold. Thus the lesser circle s,
gold, occurs only in a particular place in the larger G t
yellow, and intersects it without lying wholly within it ; G

LOGIC, Vou I. E

50 THE THEORY OF THE CONCEPT. [Bookl.

is similarly intersected in other places by the circles of other
yellow objects, and they all remain partially outside it.

30. Starting from the universal S, which was the rule for
s\ .r 2 , j- 3 , the original objects of comparison, we were able to
mount to higher and higher universals T, U, V, W. In
natural history, where such a series is of value, its several
members in an ascending scale have been named species,
genus, family, order, class : there is however a difference of
opinion as to what functions a universal concept must per-
form in order to represent even a species or a genus, and
the other names are applied still more divergently, and
always from points of view depending for their justification
on the special nature of the subject-matter. If we dispense
with this plea, the plea from the side of the specialist for the
significance and importance of these distinctions, the only
way to give some sort of fixed logical value to species and
genus is as follows. The only thing which suggests to the
natural mind to look for a universal, is the comparison of
individual instances which are not identical but similar.
To seek for a concept which included under it cucumbers
and mathematical principles, could only be an ingenious
joke ; but all varieties of human beings, big and little, old
and young, fat and thin, black and white, provoke the
natural mind to the search. Their sensible appearances
produce similar images, at the corresponding points of which
only such marks occur as are immediately felt to be species
of the same universal mark, such as hardness or colour;
and the relations between any two of these points are in all
cases merely modifications, differing in degree and amount,
of one and the same universal relation. The comparison of
individual men, therefore, produces a universal image ; not
indeed in the sense that the universal man can really be
painted, but in the sense of the illustrations in a natural
history, which purport by one camel or horse to exhibit all
camels or horses clearly to perception, in a form which is
more than a mere scheme or symbol ; or again in the sense

Chap. I.] THE INVERSE RA TIO. gi

of geometry, in which a drawn triangle, though necessarily
individual with others existing beside it, yet represents all
these others, and in a similarly perceptible form. But this
possibility vanishes when we ascend to higher universals, in
which these universal images are themselves included in
their turn as species : the universal mammal, which is
neither horse nor camel nor is otherwise named, cannot
even be drawn in a schematic form, any more than the
polygon can which has neither three, four, or any other
definite number of sides. Thus these higher universals are
no longer apprehended in perception, but only in thought,
by means of a formula or equation, which prescribes
essentially the same relation between various related points,
but leads to quite different perceptible configurations,
accordingly as the previously undetermined values of these
points and their various connexions are differently deter-
mined in thought. I would then call a universal which
still admits of an image, a species, and the first of those
which can only be expressed by a formula, a genus, in
agreement, as I believe, with the instinct of language, and
incidentally also with the old terminology of Aristotle ; for
in his choice of the words fldos and yei/os he was no doubt
determined by their original meanings ; flSos, the species,
which includes only individuals under it, is the common
element in the look or appearance of things, while yeW
comprehends things which differ in form, but in their pro-
cess of growth, or, if they have no growth in time, in the
regulative connexion of their parts, obey the same law and
formula.

31. It remains to consider the last of the assertions
mentioned above, that of the inverse ratio between the con-
tent and extent of concepts ; this seems to me to be untrue
where its truth would be important, and to be comparatively
unimportant where it is true. The number of marks, of
which we compose our concepts, is not infinite ; the words
of language, numerous but not innumerable, suffice to de-

E 2

52 THE THEORY OF THE CONCEPT. [Book I.

note them. It may therefore easily happen that a group of
them, say ikl, occurs in several universal concepts, 5 jTand
F", at once, without its therefore representing a higher uni-
versal containing all species of S jTand V. We may class
cherries and flesh under the group i k I of red, juicy, edible
bodies, but we shall not suppose ourselves thereby to have
arrived at a generic concept of which they deserve to be
called species. I do not say that in giving exclusive pro-
minence to such groups there is always as little sense as in
this absurd instance; we shall see later how valuable the
process may be ; it helps to show, what is often useful and
necessary, that different subjects, though otherwise quite
foreign to one another and not subsumable under any
common generic concept, are nevertheless, in consequence
of a single or a few common marks, jointly liable to certain
inevitable consequences. If then anyone chooses to go on
to call these groups of marks universal concepts, he is cer-
tainly right about the inverse ratio of their content and
extent : the fewer members there are in the group, the more
sure will it be to occur in all sorts of concepts, and again,
the greater the number of different ideas compared, the
smaller will be the group of marks in which they all agree.
Of the true universal, on the other hand, which contains the
rule for the entire formation of its species, it may rather be
said that its content is always precisely as rich, the sum of
its marks precisely as great, as that of its species themselves;
only that the universal concept, the genus, contains a num-
ber of marks in a merely indefinite and even universal
form ; these are represented in the species by definite
values or particular characterisations, and finally in the
singular concept all indefiniteness vanishes, and each uni-
versal mark of the genus is replaced by one fully determined
in quantity, individuality, and relation to others. It is true
that instances may be alleged against the universal validity
of this assertion, like that mentioned above of organic being,
to the concept of which we subordinate plants and animals ;

Chap. 1.] DIFFERENTIA, PROPERTY, STATE. 53

it may be called a logical caprice to retain the marks of sen-
sibility and motivity in this concept, with the tacit reservation
that they are both at zero in plants. But what this instance
properly shows is rather, that the higher universals, from the
genus upwards, really cease to be true universal concepts, and
pass over into groups of conditions, imposing uniform conse-
quences upon various genera, more properly so called.
The concept of organic being is such a group of marks,
ikl) which does not occur in any independent form of its
own, but in the genera in which it does occur, plants and
animals, gives rise necessarily to the same results.

32. By the preceding remarks I neither hope nor aspire
to bring about a permanent change in the traditional ter-
minology : they were intended merely as helps to a clearer
insight into the structure of concepts in general. With the
same object I add the following. I express the genus G,
so far as its concept gives the rule of combination for a
number of individual marks A B C, etc., by F [A B C\
and I assume that each of the marks admits of particular
forms, which we may call a 1 a* a* . . P & fr . . <r l <rV; also
that the principle of combination F has freedom to assume
various forms, of which we may indicate three by/J (/>, and
f. Now as the marks ABC may be of very different value
for the whole G, it is possible that the different values
assunied e.g. by A may be of decisive importance for the
configuration of the whole, and may also exercise a trans-
forming influence upon the combination of the other marks.
The consequence of this may be that, as A assumes one or
other of its values, the organisation of the whole, F, changes
from one of its particular modes to another ; the sum total
of the species of G would then be,

G=f(a l B C. ..) + < (<?B C. . . ) -f f (c?B C\
omitting for shortness' sake to express the corresponding
changes in B and C. These decisive marks, a 1 a 1 a*, are in
this case the specific differences, differentiae specificae. Thus
Aristotle, who gives them the name of Sia$o/>u, when he sub-

54 THE THEORY OF THE CONCEPT. [Book I.

ordinates man to the genus animal, usually describes the
faculty of rational thought as that peculiar characteristic, a 1 ,
of the universal psychical life, A, by which man is dis-
tinguished from all other animals ; to this we may now add,
following out what I have indicated above, that this d> not
only separates man from brutes, but also determines the
values of B and C peculiar to him, as also the mode of
their combination, i.e. the general character by which man
is distinguished from the brutes with their peculiar organisa-
tion $ or f. It may further happen that the particular
values which one or more of the generic marks, have as-
sumed in a single species, are possible in this and no other
species, and that yet they have no important influence upon
the shaping of its other marks, and do not therefore repre-
sent the nature of it in all its aspects. Such a mark is called
by Aristotle property, i8w; it is what we call a characteristic;
Aristotle gives risibility as a property of man, Hegel, in a
similar sense, the ear-lap ; both distinguished man from the
brutes, but without exhausting his nature. There are also,
according to Aristotle, marks which do riot belong to the
rigid constitution of a concept, but indicate something
which comes in contact with or happens to it ; every verb
which says, e.g. Socrates ' is sitting' or ' standing,' is an
example. Translators torment themselves in vain to find
an equivalent for both the real and the etymological sense
of Aristotle's expression crvpfie&rjKos ; what is important and
true in it answers completely to what we call state; that
this word does not nevertheless cover the usage of Aristotle
seems to me to be the fault of an inexactitude of his own,
which it is scarcely worth while to enter into. As to the
relation in fact between the concept as a whole and this
species of mark, its consideration belongs to the theory of
the judgment. In the introduction of Porphyrius to the
Aristotelian logic there is material enough for further
reflexion, though indeed of a mostly unprofitable sort,
about the likenesses and differences of the logical determi-

Chap. I.] PYRAMID OR MOUNTAIN-CHAIN. 35

nations here touched upon ; we have used them primarily
to illustrate the complex organisation of concepts, and with
this view have not always agreed with Aristotle in the form
of our exposition.

33. And now, where do we et to at last if we go on
looking for higher and higher concepts above those which
we have already found ? What form does the entire system
of our concepts assume if we suppose this task completed ?
It must be a structure resting on a broad base, formed by
all singular concepts or ideas, and growing gradually
narrower f as it rises. The ordinary view, in fact, gives it
the form of a pyramid, ending in a single apex, the all-
embracing concept of the thinkable. I cannot see much
point in this notion ; it rests entirely upon that unmeaning
subsumption under a mark, the logical value of which we
have already depreciated. A single step suffices to bring
everything at once under the head of the thinkable; we
may spare ourselves the trouble of climbing up to this
result by a pyramidal ladder ; and moreover the result itself
ignores in the most absolute and unmeaning way every-
thing which gives substance and character to thought. If
on the other hand we follow the method of subordination
to the genus, and arrange the manifold only under such
universals as still imply the notion of universally regulating
its specific conformations, we arrive not at one but at
several ultimate concepts not reducible to one another, in
which we are not surprised to recognise those very meanings
of the parts of speech which at the outset we found to be
the primary logical elements. All substantives go back to
the radical concept of something, all adjectives to that of
quality, verbs to that of becoming, and the rest to that of
relation. It is true that all these radical concepts have the
common mark of being thinkable ; but there is no common
genus over them of which their several essences form
species, nor does any one of them occupy this position in
regard to the rest ; it is not possible to apprehend some-

56 THE THEOR Y OF THE CONCEPT. [Book I.

thing as a species of becoming, or becoming as a species of
something. From this point of view the entire structure
of our concepts rises like a mountain-chain, beginning
in a broad base and ending in several sharply defined
peaks.

Transition to the form of the Judgment.

34. It was this image of a conceptual world building
itself up without a break, upon which the vision of Plato
dwelt. The first to recognise the eternal self-identity of
every concept and its significance as against the variableness
of the real world, he might well feel the charm of tracing
out all the simple elements of thought, of combining all
that could be combined, and of setting up in the organic
whole of a world of ideas the eternal pattern of which the
created world is an imperfect imitation, But neither he
nor his successors have attempted actually to execute this
essentially impossible task : still less should we now be
inclined to regard its execution as desirable. And this
not only because reality, things as they are, suggests riddles
too many and too hard to leave us any time for drawing up
an inventory of what might be but is not; for even a
perfect knowledge of the ideal world would give us little
support in understanding the real. The utmost that we
could attain by such means would be merely the image of a
fixed order, in which simple and composite concepts stood
side by side, each unchangeably self-identical and each bound
to its place in the system by invariable relations to all the
rest ; whereas what reality shows us is a changing medley
of the most manifold relations and connexions between
the matter of ideas, taking first one form and then another
without regard to their place in the system. This great
fact of change does not cease to be a fact because, in the
spirit of antiquity, we find fault with it as an imperfection
compared with the solemn rest of the world of ideas : the

Chap. I.] JUDGMENT EXPRESSES GROUND. 57

current of our thoughts is perpetually bringing it before us
again, and the mind, receiving as it does from that current
the stimulus to activity, has to exert itself to reduce even
these changeable coincidences to principles of coherence.
The next advance of logic is determined by this fact.

35. There are different considerations which lead us to
take the same step next. When new marks, of which
we were not before conscious in a concept, attach them-
selves to it without its apparently being changed, we are
directly stimulated to ask what ground can be conceived for
such a variable connexion of the two. But also when we
compare different instances of a universal, in the universal
marks of which we have already included the possibility of
many particular ones, it may still be asked on what ground
a particular mark in each instance coheres with the rest of
the content, and why this particular mark is privileged
above all the others which remain absent, though, as species
of the same universal, they might equally well be present.
Lastly, as we think of every concept as uniting a number of
marks, and these marks, though not essentially related
as members of one and the same systematic series, but
rather heterogeneous and foreign to one another, neverthe-
less determine each other and in their combination influ-
ence the accession of others, the question again recurs,
whflt is the ground of the apparent coherence in this co-
existence of heterogeneous elements. We are conscious
that when, in considering the concept, we attributed to
a certain combination of marks this position of a dominant
logical substance, operating in a number of different or
changing forms, we required and presupposed a view which
we have yet to show to be logically practicable. This then
is our present problem, either to break up these presup-
posed combinations again, or, if they can be justified,
to reconstitute them, but in a form which at the same time
expresses the ground of coherence in the matter combined.
In seeking to solve this problem, the form in which thought

58 THE THEORY OF THE CONCEPT.

will move will obviously be that of the judgment In this
a permanent conditioning member, the whole Content of
a concept, appears as subject over against the variable or
conditioned members or the sum of them, as predicates ;
the relation of the two, explaining and justifying their con-
nexion, lies in the copula, that is, in the accessory notion
which, more or less fully expressed in language, holds
together the two members of the sentence.

CHAPTER II.

The Theory of the Judgment.

Preliminary observations on the meaning and customary
division of Judgments.

IN accordance with the general plan of my exposition,
I should now have to develop the various forms of judg-
ment systematically as members of a series of intellectual
operations, each one of which leaves a part of its problem
unmastered and thereby gives rise to the next. Before
beginning this attempt, I must say a few words about other
usual modes of treatment, and my reasons for deviating
from them.

36. Every judgment formed in the natural exercise of
thought is intended to express a relation between the
matters of two ideas, not a relation of the two ideas them-
selves. Of 'course some sort of relation between the ideas
follows inevitably from the objective relation in the matter
which they represent ; but it is not this indispensable
relation in the mental media through which we endeavour
to grasp the matter of fact, but this matter of fact itself,
which is the essential meaning of the act of judgment
When we say, * gold is yellow/ it is indisputable that in this
judgment our idea of gold lies within the sphere of the idea
of yellow, and that accordingly the predicate is of wider
extent than the subject; but it was certainly not this that
we intended to express by the judgment. We wanted
to say that yellow itself belongs as a property to gold itself,
and only because this relation of fact is already presupposed

60 THE THEORY OF THE JUDGMENT. '[Book].

to exist (whatever difficulties this may involve), can it be
reproduced in a sentence in which the idea qf gold is
contained by that of yellow. Logic indeed has already
drawn attention to the fact that we are not quite right even
in making this sentence ; appealing from what we express
to what we mean, it teaches that the subject also from
its side limits the too extensive predicate; gold is not
yellow simply, but golden yellow, the rose rosy red, and
this particular rose only this particular rosy red. But even
with this correction the imperfection of this whole view
of the judgment is not mended ; for it does not , tell us
what is after all the relation between the two members so
corrected, and it loses sight entirely of the great possible
variety in the modes of their connexion. Thus gold is not
yellow in the dark ; its colour therefore only attaches to it
under a condition, that of the presence of light ; and if we
wished to connect this new experience with the previous
one in the phraseology of the view which we are now
considering, we should have to say, the idea of gold lies
simultaneously within the spheres of that which is yellow in
the light and of that which is not yellow in the dark ; but
this form of expression seems to me only to betray a
disposition to leave the really important point, the mention
of the conditional relation, and to go off upon results which
are true but quite without significance. Doubtless these
relations of extension between the ideas combined in the
judgment have their logical value ; but where the want
of them is felt, they are not so difficult but that they can be
mastered at the moment without special effort : to give
them a chief place in the consideration of the judgment
seems to me to be as erroneous as it is wearisome.

37. The technical expressions of logic 'point to the view
which I have taken here. In the judgment above the subject
in the sentence, that is, the grammatical subject, is the
word gold, the subject in the judgment, the logical subject,
is, not the idea of gold, but gold ; for it is to this only that

Chap. II.] MEANINGS OF THE COPULA. 61

yellow belongs as that which is predicated of it, and pre-
dicated in a definite sense indicated by the copula. On the
other hand, the idea of yellow is not a property of the idea
of gold in the same sense in which yellow is of gold ; the
one idea is not affirmed or predicated of the other; the
relation which exists between them is primarily no more
than this, that whenever, or whenever under certain con-
ditions, the one idea, gold, is found, there the other idea,
yellow, is also found, but that the former is not always
present when the latter is. But to explain and express
what it is which makes this relation possible, justifiable, or
necessary, is the problem of the logical judgment alone,
and it solves the problem by exhibiting through its copula
the relation between the object-matters of the two ideas, a
relation due to that which the ideas represent and differing
in different cases. On the other hand, it is only between
these object-matters that a logical copula is conceivable ;
between the ideas there is no relation but that of the
psychological connexion mentioned above, and that of the
monotonous, unmeaning inclusion of the one within the
other.

38. It is now clear that for us there can be only so many
essentially different forms of judgment as there are essentially
different meanings of the copula, that is, different accessory
notions which we form of the connexion of the subject with
its predicate, and to which we give more or less complete
expression in the syntactical form of the sentence. Thus
many other distinctions which meet us in logic have no use
or place in our systematic survey, though they may still
have a logical value of some other kind. To' secure clear-
ness in what is to follow, therefore, it is desirable to give a
preliminary explanation of traditional views ; but I think I
may confine it to that division of judgments to which Kant
has given currency in Germany, though it is itself of much
older date. According to Kant, as we know, the character
of every judgment is determined in four respects, quantity,,

62 THE THEORY OF THE JUDGMENT. [Book 1.

quality, relation, and modality, and in each respect every
judgment has necessarily one of three mutually exclusive
forms. I may exclude the third member of this division
from these preliminary considerations, for relation (between
subject and predicate), in respect of which Kant distinguishes
categorical, hypothetical, and disjunctive judgments, clearly
concerns just those essential characteristics of the judgment
which we are looking for, and which I shall have sub-
sequently to expound myself. If the categorical judgment
connects its subject *S and its predicate P absolutely ', as the
phrase is, or on the simple model of the relation oa thing
to its property, while the hypothetical assigns P to S, not
immediately, but only on the assumption that a certain
condition is fulfilled, and the disjunctive gives S no definite
predicate, but imposes on it the necessity of choosing
between several mutually exclusive ones, there is no doubt
that in each of these three forms the sense of the copula,
the mode of connexion between S and P, is different and
peculiar ; these three will form the series of judgments
which we shall have subsequently to construct ; only the
nine remaining ones call for the following preliminary
remarks.

39. In respect of their quantity judgments must be either
universal or particular or singular. If we express these
distinctions by the usual formulae, ' all S are P,' * som^ S
are Pf c this S is /V it is clear that they indicate merely
the different extents to which a connexion between S and P
is supposed to hold good ; the nature of the connexion in
all the cases is the same, and must be the same, because
the universal judgment, according to this view of its
meaning, admits of being formed by summing the singular
and particular ones, and must therefore be perfectly homo-
geneous with them. Thus the quantitative description
applies to the subject only, and has no reference to the
logical relation between it and its predicate ; it is therefore
of importance where the connexion of ideas requires the

Chap. II.] QUALITY OF JUDGMENTS. 63

application of a judgment, the import of which depends
upon the circuit over which it holds good ; but no special
advance in logical activity is indicated by these distinctions
as they are here formulated. I say ' as they are here
formulated,' because certainly the quantitative differences
of judgments are really connected with important logical
differences in the mode of connexion between S and P\
for doubtless that which belongs to all S has also a different
hold upon the nature of its subject from that which belongs
only to some; but the quantitative formulation of the
judgment, which merely counts the subjects, just fails to
seize this important accessory notion, and makes the relation
of the predicate to its subject, often in violation of the fact,
appear the same in all cases.

4O. In respect of quality Kant distinguished affirmative,
negative, and limitative judgments. Nothing is clearer than
that the two sentences '*$ is PJ 'S is not P,' so long as they
are supposed to be logically opposed to one another, must
express precisely the same connexion between S and P, only
that the truth of that connexion is affirmed by the one and
denied by the other. It is useful, though certainly not
necessary, to make this clear to ourselves by splitting each
of these judgments into two. We think of a certain relation,
whatever it may be, between S and P expressed in the judg-
ment < is P* as an idea still open to question ; this relation
forms the object-matter upon which two opposite judgments
are passed ; the affirmative gives it the predicate of validity
or reality, the negative refuses it. In the connexion of our
thoughts it is of course of the greatest importance "which of
these judgments is subsequently passed upon 'a given con-
nexion between and P\ but this difference does not give
rise to two essentially different kinds of judgment as such ;
validity or invalidity are rather to be considered, in regard
to the question before us, as predicates of fact to which the
whole content of the judgment forms the subject. This
content itself can be expressed in a form as yet neither

64 THE THEORY OF THE JUDGMENT. [Book I

affirmative nor negative in the interrogative sentence, and
this indeed would take the third place amongst the three
qualities of judgment more appropriately than the limitative
or infinite judgment, which is supposed to attribute a nega-
tive predicate to the subject by a positive copula, and is
usually expressed in the formula c S is not-/*.' Much acu-
men has been expended even in recent times in vindicating
this form of judgment, but I can only see in it an unmean-
ing product of pedantic ingenuity. Aristotle himself saw
clearly enough that such expressions as i not-man ' are no
concepts they are not even apprehensible idea". The
truth is that, if ' not-man ' means all that it ought logically
to mean, that is, everything that is not man, triangle, melan-
choly, sulphuric acid, as well as brute and angel, it is an
utterly impossible feat to hold together this chaotic mass of
the most different things in any one idea, such as could be
applied as a predicate to a subject. Every attempt to affirm
this unthinkable not-P of -5* will be found by an unsophisti-
cated mind to end in denying the thinkable P of the same
Sj instead of saying, ' spirit is not-matter,' we all say, c spirit
is not matter.' Even in cases where in natural thinking we
seem really to make a limitative judgment, as e.g. when we
say ' doctors are non-combatants/ we are in truth making
only a negative one. For this not-P has not here the
meaning which the limitative sentence would give it; ac-
cording to that, horses, wagons, triangles, and letters would
be non-combatants; what is meant is only human beings
who belong to the army but are declared to take no part in
fighting. Thus there is never any necessity to the natural
mind for forming limitative judgments ; every inference
which could be drawn from '5 is not-P' can also be drawn
from '* is not P.' It is not worth while to spend more
words on this point ; obvious vagaries in science must not
be propagated even by a too elaborate polemic.

41. Through the forms of modality different values are
supposed to be given to the relation which is conceived

Chap. II.] REAL MODALITY. 65

to hold between S and P\ the problematic judgment ex-
presses it aj merely possible, the assertorial as real, the
apodeictic as necessary. But these new properties are treated
quite independently of the way in which judgments have
been already determined from the other three points of
view. After it has been fixed whether a given judgment
J connects its elements in categorical, hypothetical, or dis-
junctive form, after it has been decided nj whether it affirms
or denies the relation conceived in one of those forms, and
after the extent of the subject to which the predicate applies
has been limited by the expression of quantity, it is still
held to be an open question whether the judgment so com-
posed will be problematic, assertorial, or apodeictic. To
treat the matter thus is to confess openly that the possibility,
reality, or necessity, spoken of here, stand in no connexion
with the logical construction of the judgment. All these
judgments, which are usually expressed in the formulae
' S may be /V ' S is P,' ' S must be P,' are entirely the
same as regards the validity which they give to their con-
tents by logical means ; they are all merely assertions of
the person who makes them, and are distinguished only by
their object-matter. This, the possibility, reality, or ne-
cessity of a relation between S and P, they express either
without any grounds at all, or upon grounds derived from
right reflexion upon the facts, which they do not then allow
to appear in any way in their logical structure ; just for this
reason they need additional auxiliary verbs, in order to
express independently what does not lie in the form of the
judgment itself. In more developed connexions of thought
such judgments of course have their value; for what is
wanted is often to compress results of previous reflexion
into the shape of simple assertions, without perpetually
repeating the grounds upon which they rest; here these
auxiliary verbs are in place, expressing in the form of a now
familiar fact the possibility, reality, and necessity which
once had a logical justification. But for the separation of

LOGIC, VOL. I. P

66 THE THEORY OF THE JUDGMENT. [Book I.

essential forms of judgment and their systematic arrange-
ment, the only modality that could be of vakie would be
one which, instead of going its own way independently of
the logical nexus of the other judgments, grew out of that
nexus itself, and expressed the claim to possible, real, or
necessary validity, which the content of the judgment de-
rives from the mode in which its elements are combined.

42. It would be useless to ask for such a modality, if we
could not show the possibility of it. I will therefore an-
ticipate somewhat what I have to say later. The proposition,
'all men must die/ is usually held to be apodeictic ; I con-
sider it merely assertorial \ for it states only, and does not
give grounds for, the necessity of which it speaks ; so far as
its form goes it does not even decide whether all men die
for the same reason, or everyone for a special reason, so
that the various conditions agree merely in the fact that
they leave no one alive. And yet what we had meant
by the sentence was, not only that all men as a matter
of fact die, but that the extension of mortality to all has
its ground in the universal concept of man, in the nature of
humanity and this thought we do in fact express by the
general form of the judgment 'man dies'; for the sense
of this judgment, the difference of which from the ordinary
universal I shall come back to, is not of course that the
universal concept man dies, but that everything dice which
is included under it, and for the reason that it is so in-
cluded. Every hypothetical judgment, again, gives in its
protasis the ground for what is stated in its apodosis,
and is therefore in my sense an apodeictic form of judg-
ment j the apodosis here is not simply asserted, but asserted
conditionally upon the validity of the protasis ; but, pre-
supposing that validity, the content of the apodosis is
no longer a mere fact, but a necessity, with the same right
with which every consequence necessarily follows from its
conditions. Similar remarks might be made, if they would
not be too long for this preliminary section, about the

Chap. II.] VALUE OF JUDGMENT-FORMS. 67

disjunctive judgment; and thus we should have found in
the three forms of relation three forms also of apodeictic
modality.

43. I will guard myself against a misunderstanding, though
it would be so gross that I am almost ashamed to do so.
The form which we give to the content of a judgment
can never guarantee its truth to fact ; this always depends
upon whether the relations between the elements of the
content itself are truly such as the form of the judgment, in
order to ascribe to them a certain sort of validity, has
to presuppose. This holds good of the ordinary modality
no less than of that which we would put in its place. In the
ordinary form of the apodeictic judgment, ' S must be /*,'
any nonsense may be expressed without thereby becoming
sense ; and it is equally open to us to misuse the judgments
which I call formally apodeictic, and say l man is omni-
potent/ ' if it rains everything is dry, 7 ' every triangle is
either curved or sweet or hasty-tempered/ These latter
forms of judgment, then, do not, any more than the former,
make every connexion of concepts which is put into them
true or necessary; the significance of them lies merely
in showing the formal conditions under which we may
ascribe demonstrative certainty to a given content, if that
content is in itself such as to satisfy them. And here our
view of modality differs to its advantage from the ordinary
one. The latter merely tells us that there is demonstrative
knowledge, and that, if we have got it, we can express it in
the form ' S must be P'\ but it does not tell us how know-
ledge must look, and what its internal structure ,must be, in
order to be demonstrative and to justify this expression.
Our plan on the other hand does show us this ; we find
that there are three forms of relation between *$ and P,
which, when they, exist, lead to necessary knowledge ;
endeavour to bring your ideas into one of these forms ;
either frame general judgments and look for the P which is
already implied in the conception of a genus S ; this P then

F 2

68 THE THEORY OF THE JUDGMENT. [Book I.

belongs necessarily to every species of S: or form hypo-
thetical judgments, and show that the addition to of
a condition X gives rise to a P which would not otherwise
be present ; then this P holds necessarily of every S which
comes under the same operation of the same conditions :
or lastly form disjunctive judgments ; as soon as you have
brought a question to a definite ' either . . or/ the thing
is settled, and all that is now wanted is experience to deter-
mine, in each particular instance, which of two predicates,
P or (?, will be true and necessarily true. There are no
other ways of arriving at necessary knowledge, and every
judgment which we express in the form, 1 S must be PJ
remains merely an assertion, the matter of which, if it is
convincing, has always been originally apprehended in one
of those three ways.

44. Thus far I have spoken only of apodeictic judg-
ments : the ambiguity of the ordinary theory of modality is
still more striking in the case of problematic judgments.
The proposition, 'all bodies can be set in motion by
adequate forces/ may have any one of the three modalities
ascribed to it with about equal right. Firstly, as a state-
ment which does not add the grounds upon which it is
made, it is assertorial : but what it states is not a real
occurrence, but the possibility of an unreal or only con-
ceived one, and this is enough according to traditional
usage to give it the name of problematic : lastly, it may be
called apodeictic, because it ascribes a property to all
bodies, a property therefore which can be wanting in none
and is accordingly necessary to each : in fact, this judgment
contains the reality of the necessity of a possibility. From
which point of view are we to choose its name ? I should
be in favour of regarding it as an assertorial judgment,
reckoning the necessary possibility as part of the matter
asserted. As however the same view may be extended to
all problematic judgments of the ordinary form, the ques-
tion arises whether there is any form of judgment at all

Chap. II.] QUESTION AND PRAYER. 69

which, as such, deserves to be called problematic. Inter-
rogations and prayers have been alleged as instances, for
neither of them really asserts anything ; the connexion of *S
and P which forms their content seems to be presented to
the mind as no more than a floating possibility. I doubt
however whether they can be considered as specific logical
forms at all. For ultimately interrogation must be dis-
tinguished from prayer, and the distinction can only lie in
the fact that the conscious attitude of the questioner to his
question is different from that of the petitioner to his peti-
tion. Suppose the import of the question to be, * I do not
know whether S is P* and that of the petition, ' I wish that
S were P" ; it would of course be very pedantic to say that
the speaker himself must always analyse what he says into
this bipartite form, but still, if we take his consciousness as
a whole, it must contain in both cases two different states,
tempers, dispositions, or whatever we call them, which, if
we wished to express them, could only be expressed in
those ways. If this is so, it is clear that both judgments
contain a principal sentence of an assertorial form, which
says nothing about the content of the judgment but merely
indicates the attitude of the speaker to it ; the other and de-
pendent sentence, introduced by the conjunctions ' whether '
or ' that,' comprises the whole content, without saying any-
thing about the nature and degree of its validity. It is for
this reason that I do not consider the dependent sentence
either to be a problematic judgment ; for it is not enough
that the account of the nature of the import should be
merely absent ; the import ought to be explicitly confined
to mere possibility. As to the prayer, it might further be
said that it contains the possibility of what is prayed for
and nothing else, whereas the question, as it may be a
question about possibility itself, does not always do even
that : in both moreover the assumption of the possibility of
a conceived connexion between S and P could only be
reckoned as a state of the speaker's mind, and would not

70 THE THEOR Y OF THE JUDGMENT. [Book I.

lie in the logical form of the judgment. I should rather
consider this dependent sentence to express ^without any
modality the mere content of a judgment ; and it is just
because no complete judgment can be expressed without
claiming possibility, reality, or necessity for its import, that
these sentences void of modality never occur indepen-
dently, but are always governed by some other independent
sentence which asserts one of those modalities of its con-
tent.

45. According to our view those judgments only could
be called problematic which by their logical forpn charac-
terise a conceived relation between S and P as possible
and only as possible. This is done by all quantitatively
particular and singular judgments. All that is directly ex-
pressed by sentences of the form, * Some are /V ' Some
S may or must be P,' l This S is P' or ' may or must be
P' is the actual, possible, or necessary occurrence of P in
certain cases of S -, they leave it doubtful how the matter
stands with the other cases of S which are not mentioned ;
for S as such, therefore, it is only the possibility of each of
these three relations to P which is expressed, and these
particular sentences are equivalent to the assertions, ' <S may
be P possibly/ * S may be P] * S may be P necessarily.' I
therefore call particular sentences problematic in respect of
the universal *$"; the fact that they are clearly also assertorial
in respect of the some S of which each speaks, does not at
all militate against my view ; it only shows us that in fact
the only way of recognising a certain relation between -5*
and P to be merely possible is by observing that the rela-
tion does, may, or must hold good of some S and not of
others. There are therefore certainly no independent prob-
lematic judgments, which are not assertorial in respect of a
part of their universal subject in so far as they affirm of it a
possible, actual, or necessary predicate.

46. Lastly, it is easy to see that, on the one hand, the
1 may' and * must' of the ordinary problematic and apodeictic

Chap. II.] VARIOUS MODALITIES. 71

judgments and the 'is' of the assertorial by no means
suffice to express all material differences of importance in
the truth of their several -contents, and that on the other
hand, just for this reason, they lump together very different
relations under the same expression. Firstly, what modality
have such sentences as these, * S will be P,' ' S ought to
be jP,' ' S may be P,' ' S has been P' ? No one of them
affirms reality, but the unreal which is past in the last is
something quite different from that which is permitted,
enjoined, or future, in the others : in the third it is possible,
in the second its possibility is doubtful, in the first its reality
is inevitable, while in the last it is at once irretrievable and
unreal. If all these shades of meaning had been taken into
account, the forms of modality might have been correspond-
ingly increased in number. On the other hand, how entirely
different in meaning are the similarly formed sentences,
'It can rain to-day V 'The parrot can talk/ 'Every quad-
rangle can be divided into two triangles.' In the first case
we have a supposition which is possible because we know
no reason to the contrary ; next a capacity which exists
upon conditions which need not have existed ; lastly a
necessary result of an operation which we may carry out or
not as we please. I will not multiply these instances, as
might be done indefinitely ; to attempt to analyse them all
would be as foolish as to undertake to work out beforehand
all possible examples in a mathematical text-book. In
practice, indeed, it is just from these material varieties of
meaning in the expressions in question that our inferences
are drawn but we have no resource except to observe in
each particular instance what we have before, us ; whether
it is a possibility which may be tentatively assumed in the
absence of proof to the contrary, or a well-grounded
capacity resting securely upon its conditions ; whether it is

1 [' Es kann hente regnen ; der Papagei kann reden ' ; in English we
say, ' It may rain to-day,' so that the difference of meaning is represented
by some difference of form.]

72 THE THEORY OF THE JUDGMENT. [BookU

a necessity due to the presence of imperative reasons, or
one arising from a command, a purpose, a duty, or lastly
one of those combinations of possibility, reality, and ne-
cessity which we touched upon above.

The series of the forms of Judgment.

A. The Impersonal Judgment, The Categorical Judgment,
The Principle of Identity.

47. There can be no doubt that in the series of the forms
of judgment the categorical comes before the hypothetical
and the disjunctive. We could have no- occasion for
making the occurrence of a predicate P in a subject S
dependent on a previously fulfilled condition, unless we
had already had experiences of the presence of P in some
S and its absence in others. Equally little can we think of
prescribing to S the necessary choice between different
predicates, until previous experiences have established the
constant relation of S to a more universal predicate, of
which the proposed alternatives are specific forms ; and
these experiences too would find their natural expression in
a judgment of the form ' S is P.' The structure, moreover,
of the hypothetical and disjunctive judgments exhibits per-
manent traces of this dependence : however complex they
may be in particular cases, the general scheme to which
they are reducible is always that of two judgments of the
form ' S is PJ combined, either as protasis and apodosis or
as mutually conclusive members, so as to form a single
complete assertion. But the question may be raised
whether a still simpler form must not precede the categorical
judgment itself in the systematic series. The sentence
'AS* is P' cannot be uttered until the current of ideas has
informed us of an 5 with a fixed position and recognisable
character of its own, to which a P can be added in thought
as a predicate. Now this will not always be the case ;

Chap. II.] THE IMPERSONAL JUDGMENT. 73

indeed it may be questioned whether the discovery of the
definite 5, which is to serve as subject to a categorical
judgment, does not always presuppose experiences of S in
a less developed form, and their translation into logical
equivalents. This question, which relates to the psycho-
logical growth of thought, I leave unanswered here ; for our
present purpose the fact is enough that even our fully
developed thought has preserved a form of judgment which
performs this simplest of functions, that of giving logical
setting to a matter of perception without regarding it as a
modification or determination of an already fixed subject.
This is the impersonal judgment, which, as the first act of
judging, I here treat as a preliminary stage to the cate-
gorical.

48. I do not think it necessary to defend at length the
logical import of the impersonal judgment against the
opinion which would make it merely the linguistic expres-
sion of perception itself, without involving any logical
activity. The natural sound which a man who is shivering
with cold makes when he cowers against another, is a mere
sign of this sort, which only serves to give tongue to his
feeling ; but as soon as he expresses his discomfort in the
sentence * it is cold,' he has undoubtedly performed an act
of thought. By giving to the content of his perception,
which in itself is undivided, this bipartite form of a predi-
cate related to a subject by a copula, he expresses that he
can think of it as a perceived reality in no other form
than this. It is true that he is not in a position to give the
subject an independent content; he only indicates its
empty place and the fact that it requires filling, either by
the indefinite pronoun, or in other languages by the third
person of the verb, which he uses instead of the infinitive :
it is true also that the whole content of the perception
which he expresses falls into the predicate alone : and it is
true, lastly, that the copula which he puts between them
has not as yet the sense of a definitely expressible relation ;

74 THE THEORY OF THE JUDGMENT. [Book I.

it only keeps formally apart what is substantially inseparable
and interfused. But it is just by this attempt to bring
about an articulation to which the matter of perception will
not yet lend itself, that the impersonal judgment expresses
all the more clearly the instinct of thought, that everything
which is to be matter of perception must be conceived as a
predicate of a known or unknown subject.

49. I will now explain why I have here spoken repeat-
edly of perception 1 . The indefiniteness of the subject in
the impersonal judgment has been interpreted to mean
that it merely expresses in substantival form what is
expressed in verbal form by the predicate. I do not
doubt that anyone who is asked what he means by 'it,'
when he says ' it rains/ or l it thunders,' can easily be
driven to say, ' the rain rains,' or ' the thunder thunders.'
But I believe that in that case his embarrassment makes
him say something different from what he really intended
by his impersonal judgment. It seems to me to lie in
the essence of such a judgment that he really looks upon
the determinate matter in question as attaching to an in-
determinate subject, the extent of which is much wider
than that of the predicate; and if he uses several such
expressions one after another, * it lightens,' ' it rains,' * it is
cold,' though he does not expressly intend to say that the
indefinite pronoun means the same in all those cases, he
would certainly, if he understood himself correctly, give this
answer rather than the former one. This 'it' is in fact
thought of as the common subject, to which the various
phenomena attach as predicates or from which they pro-
ceed , it indicates the all-embracing thought of reality,
which takes now one shape, now another. This has been
rightly felt by those who found in the impersonal judgment
a judgment of existence, and transformed the sentence 'it
lightens' into 'the lightning is.' It is only the transforma-
tion itself which seems to me unnatural ; we never express
1 [' Wahrnehmung.']

Chap. II.] CATEGORICAL ASSERTION. 75

ourselves in this way ; the unsophisticated mind does not
think of the phenomenon as if it were already something
before it existed, of which we could speak, and of which
among other things we could assert reality; on the contrary,
it regards the particular reality in question as a phenomenon,
a predicate, a consequence, proceeding along with others
from an antecedent and permanent, though quite inexpres-
sible, subject. Though however we cannot accept this
explanation, it is so far right as that every genuine imper-
sonal judgment expresses an actually present perception,
and is therefore as regards its form an assertorial judgment.
Such genuine judgments are to be distinguished from other
modes of expression which begin with the indefinite 'it 7 as
subject, but immediately fix its content by an explanatory
sentence, as, e. g. ' it is well that this or that should be done.'

60. The more definitely the mind emphasizes the neces-
sity of the subject to which the predicate is to attach, the
less can it rest content with an expression in which this
demand is unsatisfied. It is not part of my logical task, as
I have already said, to describe the processes of comparison
and observation by which our ideas of those subjects are
gradually formed, which we require to take the places of
the indefinite 'it' in the various impersonal judgments;
I have only to point out the logical form in which this
requirement is satisfied. Most of the simple instances with
which logic usually begins its illustration of the judgment
in general, are in the familiar form of the categorical judg-
ment ' S is /y e.g. 'gold is heavy,' 'the tree is green/
4 the day is windy.' No explanation is needed as regards
this form ; its structure is perfectly transparent 1 and simple :
all that we have to show is, that this apparent clearness
conceals a complete enigma, and that the obscurity in
which the sense of the copula in the categorical judgment
is involved will form a motive that will carry us a long way
in our successive modifications of logical activity.

61. A certain embarrassment is at once observable as

76 THE THEORY OF THE JUDGMENT. [Book I.

soon as we ask in what sense 6* and P are connected in the
categorical as distinct from the hypothetical an d, disjunctive
judgments. A common answer is, that the categorical
judgment asserts of P absolutely, but this answer is only
negatively satisfactory, i. e. so far as it denies of the cate-
gorical sentence the idea of a condition and the idea of an
opposition between mutually exclusive predicates ; but when
we know what this form of judgment does not do, the state-
ment that it joins P to S absolutely gives us no positive
information as to what it does do.- Such a statement in
fact merely expresses the greater simplicity of the cate-
gorical copula as compared with that of the hypothetical
and disjunctive judgments; but this simpler connexion
must still have a determinate and expressible sense of its
own, distinguishing it from other conceivable forms of
connexion equally simple or more complicated. The ne-
cessity of explaining this sense appears most simply from
the fact, that, of all connexions of S and P, the complete
identity of the two would be that which most obviously
deserved the name of absolute. Yet it is just this which
as a rule is not intended in the categorical judgment :
'gold is heavy' does not mean that gold and weight are
identical; equally little do such sentences as 'the tree
is green,' 'the sky is blue,' identify the tree with green
and the sky with blue. On the contrary, we are at oains
to express our real meaning in such judgments by saying,
' P is not S itself, but only a predicate of ,' or * S is
not P^ it only has P? We thus admit that we are thinking
of a definite and distinguishable relationship between S
and P, and it only remains to make really clear what
constitutes this 'having' which we oppose to 'being/ or,
in more logical language, wherein we have to look for
that relation of a subject to its predicate which we wish to
distinguish from the relation of identity.

52. Plato was the first to touch this problem; his
doctrine, that things owe their properties to participation

Chap. II.] PREDICATION NO REAL RELATION. 77

in the eternal universal concepts of those properties, was
rather an' inadequate answer to a metaphysical question
about the structure of reality, than an explanation of what
we have in our mind when we establish a logical relation
between subject and predicate. Aristotle made the right
treatment of the question possible by observing that the
attributes are primarily enunciated of their subjects; this
at any rate established the fact that it is a logical operation
of the mind which refers the matter of the one concept
to that of the other ; but more than this name of enun-
ciation, "arqyopeu/, from which that of the l categorical'
judgment and that of the Latin equivalent * predicate'
are derived, even Aristotle did not discover. He escaped
indeed a confusion of later logic; he did not reduce
the connexion which he supposed between S and P from
a logical operation to a mere psychical occurrence, thus
making the relation between the two consist only in the
fact that the idea of P is associated in our consciousness
with that of S: for him the sense of the judgment and
the ground for making it was a real relation between the
matters of the two ideas. But he did not tell us how
precisely S is affected by the fact that we enunciate P
of it; he made the enunciation itself, which can really
do nothing but recognise and express this real relation,
stand for the very relation which it had to recognise.
Now it is easy to see that this fusion is quite inadmissible ;
it is impossible merely to enunciate the concept * slave' of
Socrates in such a way that the enunciation itself should
settle the relationship in which the two concepts stand
to one another: what we really mean by a judgment is
always, that Socrates is or is not a slave, has or has not
slaves, liberates or does not liberate slaves. It is one or
other of these possible relations which constitutes what
is enunciated in each case, and it is only a matter of
linguistic usage that, when we speak of enunciating the
latter concept of the former, we choose tacitly to under-

7 8 THE THEORY OF THE JUDGMENT. [Book I.

stand only the first relation, viz. that Socrates is a slave.
The relation, therefore, of 6" to P in a categorical judgment
is not distinguished from other relations by saying that
P is enunciated of ; the truth rather is that the meaning
of this enunciation, in itself manifold, is determined by
the tacit supposition that P is enunciated of S as a predicate
of a subject. It still remains a further question, what con-
stitutes this peculiar relation.

53. We moderns are accustomed on this point to hold
to the doctrine of Kant, who represented the relation of
a thing to its property, or of substance to its* accident,
as the model upon which the mind connects S and P in
the categorical judgment. This statement may have a
good meaning in the connexion in which Kant made it,
but it does not seem to be available for the logical question
before us. I will not here raise the point whether the
idea of the relation between substance and attribute is
itself so clear and intelligible as to dissipate all obscurity
from the categorical judgment; it is enough to remind
ourselves that logical judgments do not speak only of
what is real, of things ; many of them have for their subject
a mere matter of thought, something unreal, or even im-
possible. The relation existing between the real thing
as such and its properties obviously cannot be transferred
in its full sense to the relation of subjects to their pre-
dicates, but only in the metaphorical or, as we may say,
symbolical sense. To speak more exactly, the only common
element in these two kinds of relation is the formal one,
that in both the one of the related members, thing, or
subject, is apprehended as independent, the other, property
or predicate, as dependent upon the former in the way
of attachment or inherence. But in regard to the thing,
metaphysic has at any rate exerted itself to show how
there can be properties which are not the thing and yet
attach to it, and what we are to suppose this attachment
to consist in; whereas in regard to the relation between

Chap. II.] PRINCIPLE OF IDENTITY. 79

subject and predicate we find no corresponding account
of the sense in which the one inheres in the other. The
appeal to the relation between thing and property, there-
fore, does not help logic at all ; the question repeats itself,
How much of this metaphysical relation survives as a
logical relation expressible in the categorical judgment,
if the thing be replaced by something which is not a thing,
and the property by something which is not a property ?

54. Without adding any more to these customary but
unsuccessful attempts to justify the categorical judgment,
I will state the conclusion to which we are driven : this
absolute connexion of two concepts S and P 9 in which the
one is unconditionally the other and yet both stand over
against each other as different, is a relation quite imprac-
ticable in thought ; by means of this copula, the simple ' is '
of the categorical judgment, two different contents cannot
be connected at all; they must either fall entirely within
one another, or they must remain entirely separate, and the
impossible judgment S is P ' resolves itself into the three
others, 'S is S,' 'P is P,' ( S is not P.' We must not stumble
too much at the startling character of this assertion. Our
minds are so constantly making categorical judgments of
the form ' S is PJ that no doubt what we mean by them
will eventually justify itself, and we shall soon see how this
is possible. But the categorical judgment requires such a
justification ; taken just as it stands it is a contradictory and
self-destructive form of expression, in which the mind either
represents as solved a hitherto unsolved problem, the
determination of the relation between S and P, or so
abbreviates the discovered solution that their connexion is
no longer visible. On the other hand we are met by the
consciousness that all our thought is subject to a limitation
or has to conform to a law ; by the conviction that in the
categorical judgment each constituent can only be conceived
as self-same. This primary law of thought, the principle of
identity, we express positively in the formula A~A, while

8o THE THEORY OF THE JUDGMENT. (Book I.

in the negative formula, A does not = non-J, it appears
as the principle of contradiction to every attempt to make
A=B.

56. I will not interrupt my exposition here by remarks
which would have to be repeated later upon the various
interpretations which this first law of thought has received ;
I will confine myself to stating exactly what sense I shall
myself attribute to it in opposition to many of those inter-
pretations. In the case of an ultimate principle, which
limits the whole of our thinking, it is obvious that with the
application of thought to different groups of objects it will
be transformed into a number of special principles, which
exhibit its general import in the particular forms in which it
applies to the particular characteristics of those groups and
has an important bearing upon them. The consequences
thus drawn from the principle of identity, some of which
are quite unexceptionable while others are by no means so,
must be distinguished from the original sense of the principle
itself, and do not belong to this part of logic. Thus it is
quite useless to expand the expression of the law into the
formula, Everything can have at the same moment and in
the same part of its whole self only one predicate A, and
cannot have at the same time a predicate non-A contrary or
contradictory to A, This statement is certainly correct, but
it is no more than a particular application of the principle
to subjects which have the reality of things, are composed
of partsj and are capable of change in time. On the other
hand it is incorrect to distinguish, as is often done tacitly
and not less often explicitly in formulating this principle,
between consistent predicates, which can belong at the same
time to the same subject, and others which cannot because
they are inconsistent with one another and with the nature
of the subject. In the applications of thought, of course,
this distinction too has its validity, when it has justified
itself before the law of identity ; but, taken as it stands,
that law knows nothing of predicates which, though different

Chap. II.] IDENTITY IN JUDGMENT. 8 1

from S, are still so far consistent with it that they could be
combined with it in a categorical judgment ; on the con-
trary, every predicate P which differs in any way whatever
from *$*, however friendly to *$* it might otherwise be con-
ceived to be, is entirely irreconcileable with it ; every judg-
ment of the form, ' S is P] is impossible, and in the strictest
sense we cannot get further than saying, *S is S' and
' P is P? The same interpretation of the principle must
also be maintained against other metaphysical inferences
which are drawn from it. It may be that in the course of
metaphysical enquiry it becomes necessary to make such
assertions as, What is contradictory cannot be real, What is
must be unchangeable, and the like : but the logical law of
identity says only, What is contradictory is contradictory,
What is is, What is changeable is changeable : all such
judgments as make one of these concepts the predicate of
another require a further special explanation.

B. The Particular Judgment. The Hypothetical Judgment.
The Principle of sufficient Reason.

56. It would be wearisome to stay longer at a point of
view in which we could never permanently rest : we will
follow thought to the new forms in which it tries to bring
its categorical judgments into harmony with the law of
identity. Judgments of the form 'S is P y are called syn-
thetical, when P is understood to be a mark not already
contained in that group of marks which enables us to
conceive S distinctly; they are called analytical when P,
though not identical with the whole of 6", yet belongs
essentially to those marks the union of which is necessary
to make the concept of S complete. In the analytical
judgment people have found no difficulty; but the syn-
thetical attracted attention at an early period, and Kant's
treatment of it in particular has recently made it con-
spicuous. He too however was mainly interested in ac-
counting for the possibility of synthetical judgments a priori^

LOGIC, VOL. I. G

82 THE THEORY OF THE JUDGMENT. [Book I.

i.e. such as assert an existing and necessary connexion
between S and a concept P not indispensable to S, without
the need of appealing to the experience of its actual
occurrence : as to synthetic judgments a posteriori, which
merely state that such a connexion between two not
mutually indispensable concepts is found or has been
found in experience, he regarded them as simple expressions
of facts and therefore free from difficulty. These distinctions
may be fully justified within the circle of enquiry in which
Kant moved ; but our logical question as to the possibility
of categorical judgments extends to all three forms with
equal urgency. The necessity of justification before the
principle of identity is only more obvious in the case of
the a priori synthetical judgment, which formally con-
tradicts that principle \ but it holds good of the a posteriori
also. For a judgment does not simply reproduce the fact
like a mirror; it always introduces into the observed
elements of the fact the thought of an inner relation,
which is not included in the observation. Experience
shows us only that S and P are together \ but that they
are inwardly connected, as we imply when we predicate P
of S in the judgment, is only the interpretation which our
mind puts upon the fact. How this relation can subsist
between subject and predicate in general, and between
S and P in particular, is just as obscure after experience
has shown the coexistence to be a fact as when we assert it
in anticipation of experience. Lastly, analytical judgments
raise the same difficulty. However much yellow may be
already contained in the concept of gold, the judgment
1 gold is yellow' does not assert merely that the idea of
yellow lies in the idea of gold, but ascribes yellowness to
gold as its property; gold must therefore have a determinate
relation to it, which is not the relation of identity. This
relation has to be explained, and the question still remains,
What right have we to assign to S a P which is not S, as
a predicate in a categorical judgment ?

Chap, li.] THE PARTICULAR JUDGMENT. 83

57. The only answer can be, that we have no right : the
numberless categorical judgments of this form which we
make in daily life can only be justified by showing that
they mean something quite different from what they say,
and that, if we emphasize what they mean, they are in fact
identical judgments in the full sense required by the prin-
ciple of identity. The first form in which we get a hint of
this in the natural course of thought is that of quantitative
judgments in general, which I shall in future call shortly
particular, and consider as the first form of this second
group of judgments. Under this title I include not only
the traditional forms, such as, ' all S are /y ' some 8 are
/y this 'S is PJ which have for their subject a number of
instances of the general concept 6', but those also which in
various other ways limit to definite cases, and therefore
particularise, the universal application of the connexion
between S and P, whether by particles of time (now, often,
etc.), or by those of space (here, there, etc.), or again by
a past or future tense of the verb, or lastly by any kind of
accessory idea, imperfectly expressed or not expressed at all.
In the general formula of the categorical judgment, ' S is
/y it looks as if the universal *S" were the subject, the
universal P its predicate, and the constant, unchangeable,
and unlimited connexion of S and P the import of the
whole judgment. If on the other hand we supply explicitly
what is suggested, or at any rate is meant, by these par-
ticularising accessory ideas, we find that the true subject is
not the universal S, but S, a determinate instance of it;
that the true predicate is not the universal P y but n, a
particular modification of it; and lastly that the relation
asserted is not between S and P, but between 2 and n,
and that this relation, if the supplementary ideas are correct,
is no longer a synthetical, nor even an analytical one, but
simply one of identity. A few instances will make this
clear.

58. We say, * some men are black/ and suppose ourselves

G 2

84 THE THEORY OF THE JUDGMENT. [Book I

to be making a synthetical judgment, because blackness is
not contained in the concept of man. But the* true subject
of this sentence is not the universal concept * man ' (for it is
not that which is black), but certain individual men ; these
individuals, however, though they are expressed as merely
an indefinite portion of the whole of humanity, are yet by
no means understood to be such an indefinite portion ; for
it is not left to our choice what individuals we will take out
of the whole mass of men ; our selection, which makes them
* some' men, does not make them black if they are not so
without it; we have, then, to choose those men, and we
mean all along only those men, who are black, in short,
negroes ; these are the true subject of the judgment. That
the predicate is not meant in its universality, that on the
contrary only the particular black is meant which is found
on human bodies, is at once clear, and I shall follow out
this remark later ; here I will only observe that it is merely
want of inflexion in the German expression which deceives
us as to its proper sense ; the Latin ' nonnulli homines sunt
nigri' shows at once by number and gender that * homines'
has to be supplied to ' nigri.' The full sense, then, of the
judgment is, ' some men, by whom however we are only to
understand black men, are black men '; as regards its matter
it is perfectly identical, and as regards its form it is only syn-
thetical because one and the same subject is expressed from
two different points of view, as black men in the predicate, as
a fragment of all men in the subject. Again, we say, * the
dog drinks.' But the universal dog does not drink ; only a
single definite dog, or many, or all single dogs, are the
subject of the sentence. In the predicate too we mean
something different from what we express : we do not think
of the dog as a sort of ever-running syphon ; he does not
drink simply, always, and unceasingly, but now and then.
And this 'now and then' also, though expressed as an
indefinite number of moments, is not so meant; the dog
drinks only at definite moments, when he 'is thirsty or at

Chap. II ] LIMITATION OF SUBJECT. 85

any rate inclined, when he finds something to drink, when
nobody stops him \ in short, the dog which we mean in this
judgment is really only the drinking dog, and the same
drinking dog is also the predicate. Again, { Caesar crossed
the Rubicon '; but not the Caesar who lay in the cradle, or
was asleep, or was undecided what to do, but the Caesar
who came out of Gaul, who was awake, conscious of the
situation, and had made up his mind ; in a word, the
Caesar whom the subject of this judgment means is that
Caesar only whom the predicate characterises, the Caesar
who is crossing the Rubicon, and in no previous moment of
his life was he the subject to whom this predicate could
have been attached. It is obvious moreover to every
capacity that when he had crossed the river he could not go
on crossing it, but was across, so that in no subsequent
moment of his life either can he be the subject intended in
this judgment. I will give two more examples, which Kant
has made famous. It is said that the judgment, * a straight
line is the shortest way between two points,' is synthetical,
for neither in the concept 'straight' nor in that of t line' is
there any suggestion of longitudinal measure. But the
actual geometrical judgment does not say of a straight line
in general that it is this shortest way, but only of that one
which is included between those two points. Now this fact,
the fact that its extension is bounded by two points, (and it
is only with this qualification that it forms the true subject
of the sentence) is the ground, in this case certainly the
satisfactory ground, for assigning the predicate to it. It is
easy to see that the concept of a straight line a I between
the points a and b is perfectly identical with the concept of
the distance of the two points ; for we cannot give any
other idea of what we mean by ' distance in space^ than
this, that it is the length of the straight line* between a and
b. There is not therefore a shorter and a longer distance
between a and , but only the one distance a <, which is
always the same. On the other hand, we can speak of

86 THE THEOR Y OF THE JUDGMENT. [Book I.

shorter and longer ways between a and b\ the concept of
way implies merely any sort of progression whidi leads from
a to b ; as this requires the getting over of the difference
which separates b from #, there can be no way leading from
a to b which leaves any part of this difference not got over ;
accordingly, that the shortest of all possible ways is the
distance, i.e. the straight line between the given points, is a
judgment which, as regards its matter, is perfectly identical,
and merely regards the same object from different aspects.
Nor again can the arithmetical judgment, 7 + 5 = 12, be
synthetical because 12 is not contained in either 7 or 5 :
the complete subject does not consist in either of these
quantities singly, but in the combination of them required
by the sign of addition ; but in this combination, if the
equation is correct, the predicate must be wholly contained ;
the equation would be false if some unknown quantity had
to be added to 7 + 5 in order to produce 12. Here too,
then, we have a perfectly identical judgment as regards its
matter, and it is only synthetical formally because it exhibits
the same number 1 2 first as the sum of two other quantities,
and then as determined by its order in the simple series of
numbers, I must now add that it is impossible to express
everything satisfactorily all at once : what it really means,
and how it is possible, that thought should represent the
same matter under different forms, we shall very soon have
occasion to consider ; and subsequently it will appear that
my late remarks were not intended to charge Kant with a
logical oversight so easily detected.

59. So far our result seems to be this : categorical judg-
ments of the form '*$* is P' are admissible in practice
because they are always conceived in the sense which
we have called particular, and as such are ultimately
identical. No % one however will feel satisfied with this
conclusion : it will be rightly objected that it does away
with the essential character of a judgment, which is that
it expresses a coherence between the contents of two ideas.

Chap. II.] IDENTITY DESTROYS JUDGMENT. 8f

In fact, if, by the supplementary additions which we spoke
of, we make our examples into identical judgments, and
thus compress their whole content into their subjects, so
that A means the black man, B the drinking dog, C Caesar
crossing the Rubicon, all that they say, except the barren
truth that A A, B B, C C, is reduced to this, that
A exists as a fact continually, B sometimes, and C has
occurred once in history. In other words, these judgments
no longer assert any mutual relation between the parts
of their content^ but only that this content as a composite
whole is a more or less widely extended fact, and this is
clearly a relapse to the imperfect stage of the impersonal
judgment. The following consideration will make us still
more sensible of this defect. I just now described B as the
concept of the drinking dog, but properly I had no right* to
do so ; for this expression, which joins ' drinking ' in the
form of a participle to the subject ' dog,' is itself only
conceivable and admissible on the assumption that the
mark of drinking, P, which is not contained in the subject
S, can really be ascribed to that subject in a categorical
judgment, and ascribed to it in the sense of its property or
state. Now just this possibility has been done away with
by our previous explanation ; all that it is now competent
to us to do is to understand B as the coexistent sum of its
marks a b c d, and to say, this a b c d^ which according to
the principle of identity is always self-same, has a certain
reality, while another aggregate of marks, a b c e, has a
similar reality on another occasion. But we have no right
whatever to regard the common group a, b, c, as something
inwardly connected, and more connected in itself than with
the varying elements d and e, still less as something which
offers a support to these changing elements as subject to
attributes. In language, indeed, we should continue to
describe this a b c as ' dog,' a b c d as ' eating/ and a b c e as
* drinking dog ' ; but these expressions would rest upon no
logical ground ; none of our judgments could express any-

88 THE THEORY OF THE JUDGMENT. [Book

thing but simple or composite perceptions, and between
the several perceptions, or even the several parts of each
composite perception, there could be no expressible con-
nexion such as could show their mere coexistence to be due
to inner coherence.

60. Against such a complete failure in its logical purpose
the mind guards itself, by further transforming the particular
judgment in a way which may be primarily considered as
a simple denial that the material of our ideas is thus dis-
integrated into merely isolated coexistent facts. The addi-
tions by which we supplemented the subject expressed in
the categorical judgment, were the means by which we
helped that judgment to justify itself before the principle of
identity ; they are now recognised as being also the valid
ground of fact which qualifies S for assuming a predicate P,
which, so long as it stood alone, would not belong to it.
The accessory circumstances, through which S first became
the true subject 2 of a then identical judgment, appear now
as the conditions , by the operation or presence of which S is
so influenced that a P, which before was strange to it, now
fits and belongs to it consistently with the principle of
identity. It is therefore the hypothetical judgment which
takes its place as the second member in this second group
of the forms of judgment ; it is compounded of a protasis
and an apodosis, which in the simplest typical case have
the same subject S, but different predicates, in the protasis
a Q which expresses the condition accruing to S, in the
apodosis a P which expresses the mark produced in
by that condition. All hypothetical judgments with different
subjects in their two members are abbreviated expressions,
and can be reduced by easily supplied links to this original
form, ' If S is Q, S is P. 9 If it is further wished to imply
that the protasis, which as such is only problematical, is
actually true, we get the form, 'Because S is Q, S is P 9 :
and lastly, the assertion that Q is not the ground for *S*s
being P gives rise to the last form which we need mention,

Chap. II.] THE HYPOTHETICAL JUDGMENT. 89

Although S is (>, yet S is not PS Logically there is no-
thing peculiar in these two forms.

61. This short survey is quite sufficient to characterise
the external forms of the hypothetical judgment. But an
observant reader must ask at this point, what right had we
to translate those supplementary additions, to which the
true subject 2 of the then identical judgment owed its
origin, into conditions , which, by operating upon an already
existing subject *S, give a ground for the predication of P.
The principle of identity merely asserts the sameness of
everything with itself; the only relation in which it places
two different things is that of mutual exclusion. If then we
supposed various simple elements a b c p q existing together
in some real form, but without being in any way inwardly
connected, some of these elements might equally well occur
at any subsequent moment in any other combination with
any other element, and the fact of our observing a b c q
a second time would not enable us to conclude that/ must
be there too; any r or s might with equal right take its
place. On the other hand, if we make the quite general
presupposition that the totality of things thinkable and real
is not merely a sum which coexists but a whole which co-
heres, then the law of identity has wider consequences.
The same a b c q, with which p has once been found in
combination, can then according to the law of identity
never be found in combination with a non^, nor can this
abcq ever occur without its former predicate/. How such
a cohesion between different elements is conceivable, we
will leave for a moment an open question; but if it exists,
it must exist in an identical form in every recurrent instance,
and (confining ourselves to a combination of three elements)
given a b> c is the only new element which can necessarily
accrue, given a c^ b^ and given be, a\ in other words, which-
ever of these elements occurs first in any case has in the
second the sufficient and necessary condition for the possi-
bility and necessity of the accession of the third. That

90 THE THEORY OF THE JUDGMENT. [BookL

element or group of elements to which we here give the
first place, appears to us then logically as the subject ; that
which we place second, as the condition which operates
upon this subject, while the third represents the conse-
quence produced in the subject by the condition. I wish
further expressly to point out that this choice of places is
quite arbitrary, and in practice is decided by the nature of
the object and our interest in it : in itself, every element in
such a combination is a function of the rest, and we can
pass inferentially from any one to any other. It is usual to
conceive of a number of elements which frequently recur
together as a subject S, which generally signifies a thing or
permanent reality: on the other hand, a single element b,
which is absent in some observations of and present in
others, is conceived as the accessory condition Q, and a <r,
which always accompanies <, as the consequence P of which
Q is the condition. But it is obvious that we may proceed
in a different way; and in fact mechanical physics are able
to treat the single and uniform force of gravity, b or <2, as a
subject, and to investigate the various consequences, P,
which accrue to it if the bodies upon which it acts (amn
= S or amr=S l ) be regarded as different conditions to
whose influence it is liable.

62. In this way the interpretation by which we arrived at
hypothetical judgments may be said to be so far justified, as
that it has been traced back to the most general assumption
of a coherence between the various contents of thought.
To prove further than this the admissibility and truth of
that assumption itself, cannot be part of our undertaking ;
any such attempt would obviously imply what had to be
proved, for how could we show that it is permissible and
necessary to conceive the matter of experience as a web of
reasons and consequences, if we did not base this assertion
itself upon a reason of which it was the consequence ? This
idea of the coherence of the world of thought must therefore
either be apprehended with immediate certitude, as the

Chap. II.] LAW OF SUFFICIENT REASON. 91

soul of all thinking, or we must give it up and along with it
everything that depends upon it. On the other hand, we
are justified in desiring further elucidation of the possibility
and the meaning of such a coherence of different elements.
The possibility of mutual relations between what is different
is not really threatened by the principle of identity, accord-
ing to which each thing is related only to itself; for all that
this principle can affirm is the content of the thing itself; it
cannot exclude other contents which do not conflict with it.
But as regards the meaning of the coherence, we must dis-
tinguish two questions. In logic as here conceived we do
not trouble ourselves at all as to what the real process may
be through which the unknown reality, which we express
well or ill through our ideas, reacts upon itself and produces
changes in its conditions ; to reflect upon the bond of this
connexion is the function of metaphysic, and the question
should find solution in a theory of the efficient cause. Logic,
on the other hand, which includes in its consideration the
relations of the merely thinkable which has no real existence
in fact, is confined to developing the other principle, that of
sufficient reason ; it has merely to show how, from the com-
bination of two contents of thought, S and <2, the necessity
arises of thinking a third, P, and this in a definite relation
to *$"; if then it were found in actual experience that such a
union of S 1 and Q l is an accomplished fact, the particular
consequence P 1 , which according to the necessity of thought
must follow such a combination in distinction from P' 2
which could not so follow, could be inferred according to
the principle of sufficient reason ; but how it comes about
that the very P 1 , which is required by thought, occurs in
reality as well, is a question which would be left to the
metaphysical enquiries referred to.

63. The law of sufficient reason^ with which we now con-
clude as the third member and the net result of this second
group of the forms of judgment, much as it has been talked
about, has had the curious fortune never to have been, pro-

92 THE THEORY OF THE JUDGMENT. [Book I.

perly speaking, formulated, even by those who most fre-
quently appealed to it. For the ordinary injunction, that
for every statement which claims validity we must seek a
ground for its validity, forgets that we cannot seek for that
of which we do not know wherein it consists ; clearly the
first thing that has to be explained is, in what relation
reason and consequence stand to each other, and in what
sort of thing consequently we may hope to discover the
reason of another thing. I shall make my meaning clear
in the shortest way, if, on the analogy of the expression ot
the principle of identity, A = A, I at once give the formula
A + B = C as the expression of the principle of sufficient
reason, adding the following explanation. Taken by them-
selves, A only =: A, B = B\ but there is no reason why
a particular combination A 4- B, the very different sense
of which in different cases is here represented by the
sign of addition, should not be equivalent to, or identical
with, the simple content of the new concept C. If we
thus call A -h B the reason and C the consequence,
reason and consequence are completely identical, and the
one is the other ; in this case we must understand by
A + B any given subject along with the condition by
which it is influenced, and by C, not a new predicate which
is the consequence of this subject, but the subject itself in
its form as altered by the predicate. In ordinary usage
this is expressed differently. Inasmuch as, in speaking ol
real facts, the one part A is usually already given, while the
other B is a subsequent addition, it is customary to describe
the condition B^ which forms only a part of the whole
reason A -f #, as the reason in general which acts upon
the passive subject A ; by C is then usually understood
nothing but the new property conditioned by B> and this is
called the consequence ; at the same time, however, the
property is never thought of as existing on its own account,
as if in empty space, but as attaching to the subject A upon
which B was supposed to act. Ordinary usage, therefore,

Chap. II.] BASIS OF MATHEMATICS. 93

though it employs a different nomenclature, means the
same as I do. If with the idea of powder, A, we connect
the idea of the high temperature of the spark, B, and thus
substitute B for the mark of ordinary temperature in A,
then A + B really is the idea C of exploding powder, not
of explosion in general ; the ordinary usage makes the high
temperature B seem to supervene on the given subject A as
a reason from which the explosion C follows ; but of course
it conceives this consequence, not as a process which takes
place anywhere, but as an expansion of the particular
powder upon which the spark acted. It is not necessary to
continue such simple explanations any further.

64. If we consider the whole of our knowledge, we see
at once that the principle of identity cannot be its only
source. Taken alone it would isolate every judgment and
even every concept, and would not open any way to a pro-
gress from the barren self-identity of single elements of
thought to their fruitful combination with others. It is a
mistake, as is sometimes done, to represent this single
principle as 4:he basis of the truths of mathematics ; the fact
is that here too it is only the principle of sufficient reason
which helps to real discovery. From a self-identical major
premiss nothing new could flow, unless it were possible in a
number of minor premisses to give the same quantity C
innumerable equivalent forms, at one time = A + J3 y at
another = M -f N^ at another = N R ; or, to express
the same thing otherwise, unless the nature of numbers
were such that we can divide them all in innumerable ways
and compound them again in the most manifold combina-
tions ; and again, unless the nature of space were such
that every line can be inserted as a part or otherwise
coherent member in innumerable figures in the most
various positions, and that each one of the expressions for
it, which flow from these various relations, is the ground
for new and manifold consequences. I need hardly men-
tion that mechanics and physics also make the most

94 THE THEORY OF THE JUDGMENT. [Book I.

extensive use of this analysis and composition of given
facts, and that the process of thought in discovery in these
branches of knowledge rests upon operations which all
ultimately come back to the typical formula, A + B = C.
To Herbart belongs the credit of having brought within the
ken of formal logic the importance of a mode of procedure
so prominent in all scientific practice.

66. Reserving further illustrations for applied logic, I
have another remark to make about the justification of the
principle of sufficient reason itself. We were only able to
show that an extension of our knowledge is possible if
there is a principle which allows us to make A + B = C.
We might accordingly attempt to assert the validity of this
principle at once, as an immediate certitude, like the prin-
ciple of identity. This is what we have done ; still there is
a noticeable difference between the two principles. The
principle of identity expresses of every A an equality with
itself which we feel immediately to be necessary, and the
opposite of which also we feel with equal conviction to be
impossible in thought. The principle of sufficient reason
lacks this latter support ; we do not by any means feel it
impossible to suppose that, while every content of thought
is self-identical, no combination of two contents is ever
equivalent to a third. The validity of the latter principle,
therefore, is of a different kind from that of the former ; if
we call the one necessary to thought because of the im-
possibility of te opposite, the other must be considered
rather as an assumption which serves the purposes of
thought, an assumption of mutual relatedness in thinkable
matter the truth of which is guaranteed by the concentrated
impression of all experience.

I wish not to be misunderstood in this last phrase. In
the first place, I do not mean that it is a comparison of
what we experience which first leads the mind to conjecture
the validity of such a principle ; the general tendency of
the logical spirit, to exhibit the coexistent as coherent,

Chap. II.] INNER EXPERIENCE. 95

contains in itself the impulse, which, independently even
of all actual experience, would lead to the assumption of
a connexion of reasons and consequences. But that this
assumption is confirmed, that thought does come upon
such identities or equivalences between different elements
in the thinkable matter which it does not make, but
receives or finds, this is a fortunate fact, a fortunate trait in
the organisation of the thinkable world, a trait which does
really exist, but has not the same necessity for existing as
the principle of identity. It is not impossible to conceive
a world in which everything should be as incommensurable
with every other thing as sweet is with triangular, and in
which therefore there was no possibility of so holding two
different things together as to give ground for a third : it is
true that, if such a world existed, the mind would not
know what to do with it, but it would be obliged to recog-
nise it as possible according to its own judgment. I will
add further that, when I speak of a kind of empirical
confirmation of the principle of sufficient reason, I do
not mean such a confirmation as the whole of our world
of thought, already articulated in accordance with that
principle, might find in the fact that external reality, so
far as it is observable, corresponds with this articulation;
I am speaking here only of the fact that the thinkable
world, the contents of our ideas which, whatever their
source, we find in our inner experience, do really con-
form to the requirement that they should cohere as reasons
and consequences. In this stage of logic it is quite in-
different whether or not there is anything which can be
called external world or reality besides the ideas which
move within our consciousness ; like that reality, this in-
ternal world itself, with all that it contains, is not made
by thought; it is a material which thought finds in us to
work upon, and it is therefore for the logical spirit and
its tendency an object of inner experience; this, then, is
the empirical object which, by responding to the logical

g6 THE THEORY Ofi THE JUDGMENT. [Book!.

tendency and making its realisation possible, substantiates
the principle of sufficient reason, not as a necessity of
thought, but as a fact.

66. As to the nature of this responsiveness in the
world of thought (if that question is to be raised again
here), the shortest way to recall it is to observe that the
position occupied in the system by the principle of sufficient
reason, as the second law of thought, is analogous to that
of the act which 1 we placed second in treating of con-
ception. The possibility of forming general concepts de-
pended on the fact, not in itself a necessity of. thought,
that every idea is not incommensurable with every other,
but that, on the contrary, colours, tones, and shapes
group themselves in series of cognisable gradations; that
further there are oppositions of varying degree, as well
as affinities, in the world of thought, and that opposites
cancel one another; and lastly, and most important of
all, that there is a system of quantitative determinations
enabling us to compare the members of different series,
which as such stand in no mutual relation. With this
brief indication, we leave the principle of sufficient reason
as the conclusion and net result of the second group of
the forms of judgment.

C. The General Judgment. The Disjunctive Judgment.
The Dictum de omni et nullo and the Principium ex-
clusi medii.

67. It remains to determine in each particular case,
What A, combined in what form with what B^ forms the
adequate reason of what C. This question of fact logic
must leave to experience and the special sciences; but
a new question is developed which logic itself must deal
with. There would be little result from all the activity
of our mind if we were really obliged in every particular
case to renew the question to experience, What A, B, and

1 [See above, 19.]

Chap.II.] THE GENERAL JUDGMENT. 97

C in this instance cohere as reason and consequence?
There must be at any rate a principle which allows us,
when once the one truth A+It=Cis given, to apply it to
cases of which experience has not yet informed us. What
we are here looking for is easy to find, and has been already
mentioned incidentally. Whenever we regard A 4- B as the
reason of a consequence C, we necessarily conceive the
connexion of the three as a universal one ; A -f- B would
not be a condition of C, if, in a second case of its occur-
rence, some casual D instead of C might possibly be found
combined^ with it. The significance of this in its present
application is as follows : everywhere, in every subject
in which A -f B is contained as a mark along with other
marks, N O P, this A 4- B gives ground for the same con-
sequence C\ and this C will either actually occur as a
mark of S, or, if it does not occur, it can only be hindered
because the other marks, N+ O or N+P or O + P, formed
together the ground for a consequence opposed to and
destructive of C; taken by itself, without this hindrance,
the power of A -f B to condition C never loses its effect.
If now we conceive A -f- B under the title M as a universal
concept to which S is subordinate, we may give the fol-
lowing preliminary expression to the principle just discovered,
viz. that by right of pure logic and without appeal to
experience every subject may have that predicate affirmed
of it which is required by the generic concept above it.
And it is clear without further explanation that this very
idea of the subordination of the individual to the universal
is the comprehensive logical instrument, of which we avail
ourselves whenever we want to carry further the work of
thought upon the material given in experience.

68. The form of judgment, the first of this third group,
in which the mind expresses this conviction, is that of the
quantitatively undetermined proposition, in which the place
of the subject is filled simply by a universal or generic
concept M\ 'man is mortal,' 'sin is punishable.' I dis-

Locic, VOL. I. H

98 THE THEORY OF THE JUDGMENT. [BodkL

tinguish these as general judgments from the universal
ones, 'all men are mortal/ t every sin is punishable/
Although the fact contained in both forms is the same,
the logical setting of it in the two cases is quite different
The universal judgment is only a collection of many
singular judgments, the sum of whose subjects does as
a matter of fact fill up the whole extent of the universal
concept ; thus the fact that the predicate P holds good of
all M follows here only from the fact that it holds good
of every single M] it may however hold good of each
M for a special reason which has nothing to do with
the universal nature of M. Thus the universal proposition^
'all inhabitants of this town are poor,' leaves it quite
uncertain whether each inhabitant is made poor by a
particular cause, or whether the poverty arises from his
being an inhabitant of this town ; so too the universal
proposition, 'all men are mortal/ leaves it still an open
question whether, strictly speaking, they might not all live
for ever, and whether it is not merely a remarkable con-
catenation of circumstances, different in every different
case, which finally results in the fact that no one remains
alive. The general judgment on the other hand, 'man
is mortal/ asserts by its form that it lies in the character
of mankind that mortality is inseparable from every one
who partakes in it. While therefore the universpl judg-
ment merely states a universal fact, and is therefore only
assertorial, the general judgment lets the reason of its
necessary truth be seen through it, and may thus, in the
sense laid down above 1 , be called apodeictic. This dis-
tinction of the two forms of judgment will not lead to
any unheard-of discoveries ; but in comparison with the
many unprofitable distinctions which encumber logic it
deserved an incidental mention. It is scarcely necessary
to remark that in the general judgment it is not the generic
concept M, occupying the place of subject in the sentence,
1 [Above, 42.]

Chap. II.] THE DISJUNCTIVE JUDGMENT. 99

which is the true logical subject of the judgment; it is
not the universal man who is mortal, but the individual
S 1 who participates in this type, which in itself is immortal.
From this we see that the general judgment is properly
an abbreviated hypothetical judgment; in its full form it
ought to stand, ' If is M, S is P,' ' If any S is a man,
this 5 is mortal.' And this justifies us in not introducing
it in our system until after the hypothetical.

69. But it is no less clear that we must make another
step. So long as a universal generic concept M occurs
as formaHy the subject in the general judgment, the pre-
dicate P which is joined to it can only be understood
with equal universality. If we say, 'man is mortal,' the
predicate embraces all conceivable kinds of mortality, and
does not determine either the manner or the moment of
death; or if we say, ' bodies occupy space,' it remains
unexpressed with what degree of density and of resistance
each single body realises the universal property of its class.
But we saw that it is individual men and individual bodies
which are the real subjects of the general judgment; it
is therefore quite false to say that P, the mark of their
class, is a predicate of the individuals in the same universal
sense in which it is joined in thought (and that not as
a predicate) to the concept of the class; the truth is
that P can only occur in each one of these individuals
in one of the definite forms or modifications into which
the universal P can be analysed or particularised. The
mind corrects this mistake by means of the fresh assertion,
' If any S is an M^ this S is either / T or / 2 or /Y and
here/ 1 / 2 / 3 mean the different kinds of a universal mark P
which is contained in the generic concept M. This is
the familiar form of the disjunctive judgment, the second
in this third group, and one which, as such, requires no
further explanation. It is usual to mention along with it
the copulative judgment (' S is both / and q and r'\ and
the remotive judgment ('-S* is neither/ nor ^-nor r')' t but

H 2

100 THE THEORY OF THE JUDGMENT. [Book I.

in spite of the external analogy of form, neither of these
has the same logical value as the disjunctive; the first
is only a collection of positive, the second of negative,
judgments with the same subject and different predicates,
which latter are not placed in any logically important
relation to each other. The disjunctive judgment alone
expresses a special relation between its members : it gives
its subject no predicate at all, but prescribes to it the
alternative between a definite number of different pre-
dicates.

70. The thought expressed by the form of* the dis-
junctive judgment usually finds utterance in two separate
laws of thought, the Dictum de omni et nullo and the
Principium exclusi tertii inter duo contradictoria ; but the
amalgamation of them in a single third law is not only
easy but necessary. The careless formulations often given
of the first are completely false, e.g. 'What is true of
the universal is true also of the particular/ 'What is true
of the whole is true also of the parts'; on the contrary,
it is self-evident that what holds good of the universal
as such or of the whole as such, cannot 'hold good of
the individual as such or of the parts as such. The only
correct formula is, quidquid de omnibus valet valet etiam de
quibusdam et de singulis, and quidquid de nullo valet nee de
quibusdam valet nee de singulis. But this form of expression
(for the history of which see Rehnisch, Fichte's Zeitschrift,
Ixxvi, i) is as barren as it is correct ; for to hold good of
all is and means from the very first nothing else than to
hold good of each one ; if therefore anything worth saying
is to take the place of this bare tautology, the nature of the
universal concept must certainly be substituted for the mere
sum of alL But in that case the principle cannot really be
accurately expressed except in a form which means precisely
the same as the disjunctive judgment ; viz. whenever a
universal P is a mark in a universal concept M 9 one of its
modifications, /W 3 , to the exclusion of thfe rest, belongs

Chap. II.] DICTUM DE OMNI ET NULLO. 101

to every S which is a species of M\ whenever a universal
P is excluded from a concept AT, no one of the modifica-
tions of P belongs to any S which is a species of M.

71. Of this complete law of thought the ordinary ex-
pression of the dictum de omni et nullo only regards >the one
and positive part, which, as we saw, cannot by itself be
accurately expressed, the general idea, namely, that the
particular is determined by its universal : the other and
negative part, which defines the manner of this determina-
tion, the idea that the particular admits only one specific
form of its generic predicate to the exclusion of the others,
has found only a partial expression in the principle of the
excluded middle. I think that I can say what I have to
say about this most simply as follows. Suppose a subject S
subordinate to M^ and that this subordination implies that
S must choose its own predicate from amongst/ 1 /'/ 3 , the
specific forms of P, a universal mark belonging to J/, then,
if there are more than two of these forms, the affirmation of
one of them as predicate of S will involve the negation of
all the rest, but the negation of one of them will not
involve the affirmation of any particular one of the rest;
what is not / l has still an open choice between /* / 3 / 4 . To
predicates of this sort it is usual to ascribe the opposition of
contrariety. If however there are only two specific forms of
P, /* and / 2 , and S must have a specific form of P for its
predicate, then not only does the affirmation of one of them
as predicate of S involve the negation of the other, but also
the negation of the one involves the definite affirmation of
the other ; / l and / 2 are then opposed to one another
contradictorily. Thus for the line (S) 9 which must have
some direction (P\ straight (/') and crooked (/ 2 ) are contra-
dictory predicates, and so for man, whose nature it is to
have sex, are male and female : for any other subjects, of
which it was not yet established whether their concepts
contained the universal P at all, these predicates would be
only contrary; for such subjects the division of their

102 THE THEORY OF THE JUDGMENT. [Book I.

possible predicates will be always threefold, they are either
male, female, or sexless, either straight, crooked, or form-
less. Now the principle of the excluded middle asserts
nothing but what we have just remarked, that of two
predicates which are contradictory for a subject S, S always
has one to the exclusion of the other, and if it has not the
one it necessarily has the other to the exclusion of any
third. So regarded, this law is only a particular case of the
more universal law of which the disjunctive judgment is
the expression, viz. that of all contrary predicates whose
universal P is contained in the generic concept M of
a subject S, S has always one to the exclusion of the rest,
and if it has not any given one, it has only left it the choice
between the others ; this choice becomes a definite affirma-
tion when it can only fall on one member, i. e. in the
extreme case where the number of contrary predicates is
only two. Such a case, which is all that is covered by the
principle of the excluded middle, is no doubt of peculiar
importance in practice, but a system of logic can only
treat it as a particular instance of the more universal prin-
ciple, which we have already mentioned several times and
which we will briefly describe as the disjunctive law of
thought.

72. It is usual to represent this differently. From motives
which are likewise only intelligible on practical grounds,
the logical desire has arisen to omit the presupposition
to which we have adhered throughout (viz. that the given
subject S be already understood to stand in a necessary
relation to the universal predicate />), and to be allowed to
speak of two predicates which hold good as contradictories
of any subject whatever. It is soon found that this is only
possible, if the aggregate of all conceivable predicates be
divided into a definite Q and the sum of all those which
are not Q or non-<2 ; it is then certain that any subject,
whatever it may mean, must be either Q or non-@, either
straight or not-straight, for not-straight will include not only

Chap.II.3 DISPARATES AND COMP ARABLES. 103

crooked, but annoying, sweet, future, everything in short
which lies outside of straight. On this point I may repeat
what I said 1 about the limitative judgment, viz. that non-(?
is not a real idea at all, such as can be treated as predicate
of a subject ; it is only a formula expressing a mentally
impracticable task, the collection of all thinkable matter that
lies outside a given concept into a single other concept.
Moreover there is no real reason for propounding this
insoluble problem ; everything which it is wished to secure
by the affirmative predicate non-Q is secured by the
intelligibly negation of Q. I therefore consider it quite
improper to speak of contradictory concepts , i.e. concepts
which are of themselves contradictorily opposed and
therefore retain that opposition when treated as predicates
of one and the same subject, whatever that subject may be:
if we want a contradictory relation which shall hold good
universally, always, and in regard to every subject, it can
only exist between two judgments, ' is (?,' ' S is not Q. 1
Accordingly the precise expression of the principle of the
excluded middle would be, that of every precisely deter-
mined subject 6" either the affirmation or the negation of an
equally determinate predicate Q holds good, and no third
alternative is possible ; wherever it appears to be possible,
S or <2 or both have either been taken in more than one
sense or in an indefinite sense in the first instance, or their
meaning has been unconsciously or involuntarily changed
in the course of reflection.

73. I have one more observation to add. No one doubts
that the same subject can be at the same time red, sweet,
and heavy, but that it is red only when it is neither green
nor blue nor of any other colour, and that it cannot be
straight and crooked at the same time. Yet it does not
seem to me to be immediately evident that, as is sometimes
asserted, the case in which two predicates / l and / 2 are
incompatible in the same subject is just that in which they
1 [See above, 40.]

104 THE THEORY OF THE JUDGMENT. [Book I.

are contrary species of the same universal P and therefore
admit of comparison, whereas other predicates / q r are
compatible in the same subject when, as species of quite
different universals P Q R, they admit of no comparison.
On this point I venture the following reflections. Every
predicate/ 1 of a subject S must be regarded, in accordance
with what we said above and the formula A + B~C> as a
consequence of a group of marks A 1 -\-J3 l in S, which group
tends in all cases (and therefore in the case of S] to produce
the same result C l (in this case /*). If now the same S is
to have at the same time the predicate/ 2 , comparable with
/*, it is easy to understand that/ 2 must depend on a group
of marks A 2 + .Z? 2 , similarly comparable with A 1 + l , existing
side by side with the latter in S, and in all cases of its
occurrence (and therefore in the case of S] giving ground
for the result C 2 (in this case /'). But the consequence
of the very comparability of A 1 + B^ and A^ + B* must be
that, according to a new principle of the general form
A+B-C, viz. [A l + l ] + [A* + jB']=C\ their meeting in
the same subject S will' furnish the sufficient reason for a
new consequence C z , in which the two specific predicates
p l and p 2 coalesce, and which, as it must resemble both
of them, we will call /*. The only reason, therefore, why
two contrary and comparable predicates p 1 and / 2 would be
irreconcileable, is that they would always give rise to a third
and simple/ 3 ; on the other hand, two disparate and incom-
parable predicates / and r, such as sweet and warm, could
coexist permanently in *$" because there is no principle
such as (A+) + (M+N)=C enabling the two disparate
grounds A+B and M+N, on which the predicates re-
spectively depend, to produce like / l and / 2 a third and
simple predicate. I will not quarrel with those who find
the whole of this exposition superfluous ; it seems to me to
have some point, when I turn from the examples which
logic traditionally employs to others which it would do well
not to forget. When anyone says of gold that it is yellow,

Chap. II.] DISJUNCTION AND INFERENCE. '105

he has, it is true, no occasion to think of this simple
property as a product of two other imperceptible ones,
which properly speaking must have been produced separ-
ately by two conditions coexisting in gold, but could not
remain separate. But when two motive forces contrary or
even contradictory in direction act upon a material point,
that which in the previous case would have been a needless
assumption is now an actual fact] we have to conceive both
of the condition which tends to produce the motion p l and
of that which tends to produce p* as operating at the point,
and of the two motions themselves as at every moment
predicates of that point, but predicates which cannot main-
tain themselves separately but coalesce in a third / 3 , the
motion in the diagonal.

And ultimately this is seen to be true in all cases. A
crooked line may appear indifferently red or green: but
if the conditions of both appearances were operating at the
same time and with the same force, it would help us but
little to assert, on the principle of exclusion, that the image
of the line cannot have these two contrary properties; it
must present some appearance. As however these two con-
ditions are comparable and capable of forming a resultant,
a third colour will appear, the production of which will
satisfy the claims of the two conditions, but will at the
same time contain the reason why the two contrary
colours, which singly they would have produced, cannot
exist separately side by side.

74. The series of judgments concludes here by an inherent
necessity. The more definitely the disjunctive judgment pre-
scribes to its subject the choice between different predicates,
the less can this uncertainty be final ; the choice must be made.
But the decision, what p^ or / 2 belongs to *$*, cannot come
from the fact (which is thus far the only fact) that S is sub-
ordinate to M, for it is just because it is a species of M
that it is still free to choose : that decision can only come
from the specific difference by which S, as this species

io6 f THE THEORY OF THE JUDGMENT. [Book I.

of M, is distinguished from other species of it, The pro-
position < J/(and every S which is M) is PJ must therefore
have added to it a second proposition which brings to light
the specific character of S, the particular subject always
in question, and shows us what species of M it is ; and
from the union of the two propositions must arise a third,
informing us what particular modification / of the universal
P belongs to this S because it is, not only a species of M>
but this species. The form of thought which combines two
judgments so as to produce a third is, speaking generally,
inference^ and it is therefore to the exposition of, inference
that we have now to pass,

Appendix on immediate inferences.

In conformity with tradition I insert some explanations
here which would more correctly come under the head
of applied logic. Of the same subject and the same
predicate P the universal affirmative judgment, A, asserts
* All S are P] the particular affirmative, /, ' Some are PJ
the universal negative, E, ' No S is P,' and the particular
negative, O, ' Some are not P.* The question is, what
immediate inferences can be drawn from the truth or un-
truth of one of these four judgments in regard to the truth
or untruth of the other three ? From the Dictum de omni
et nullo and the principle of the excluded middle t we ob-
tain the following results.

75. Between each universal judgment and the particular
of like name there is the relation of subalternation. Going
from the universal to the particular or ad subalternatam^ we
infer the truth of the latter from that of the former, but
from the untruth of the universal we cannot infer either the
truth or the untruth of the particular. The correctness
of the first inference is obvious at once, and it only requires
the removal of a misunderstanding to make the impossibility
of the second equally so. A person who denies the uni-
versal proposition, ' all are P, } is usually led to do so by

chap, no * AD SUBALTERN ANTEM: 107

having already observed some 6" which are not P ; but he
will not have included all S in this observation. His in-
tention therefore generally is merely to deny the universal
application of the proposition to all S, while leaving its
truth in single cases of S undisputed ; and thus it is that
in ordinary speech expressions such as ' It is not true that
all S are also /Y are actually understood to admit inci-
dentally the truth of the particular proposition, * some S are
P.* Logic, on the other hand, knows nothing of these
unexpressed suggestions in the denial of the universal
proposition : it recognises merely what lies in the expressed
negation itself. But it is just this which is ambiguous.
For the asserted untruth of the proposition, 'all are PJ
is equally a fact, whether the proposition is true of only some
S or of none. So long therefore as this ambiguity is not
removed by accessory statements, we cannot infer from the
negation of the universal proposition either the truth or the
untruth of the particular.

76. Going in the opposite direction, from the particular
to the universal or ad subalternantem, we infer the un-
truth of the universal judgment from that of the particular,
but not the truth. Here, too, the first conclusion is obvious,
if we avoid the ambiguity already alluded to. A person
who denies the proposition, 'some S are P, 1 may, it is
true, ; ntend merely to deny that P is confined to some
S, and the effect of his meaning that ' not only some S are
P ' would then be to affirm the universal proposition ' all
S are P. 9 But just because this consequence would directly
imply that the particular judgment, ' some S are P 9 * also
remained true, logic cannot possibly interpret the denial
of that judgment in this way. For logic this denial means
nothing but that f there is no such thing as some -S" which
are P 9 ; and what is not even true in some cases is still less
true in all. Consequently the negation of the particular
always negates the universal too. The impossibility of the
second inference explains itself; the truth of P in the case

loS THE THEORY OF THE JUDGMENT. [Book I.

of some can never prove its truth in all S: it is only
because this unjustifiable generalisation of single obser-
vations is the commonest of logical mistakes, to which
science and culture owe most of their errors, that it is
worth while to prohibit with especial emphasis this false
inference ad subalternantem.

77. Universal judgments are contradictorily opposed to
particulars of unlike name, A to O and to I and vice
versa ; we infer ad contradictoriam both the untruth of
the one from the truth of the other and the truth of the
one from the untruth of the other. The first inference
needs no explanation, the second a brief one. If we
deny the proposition A, 'all are P,' the denial is con-
sistent with both the assumptions E and O, ' no S is P,'
and ' some are not P' ; but the second, which is included
in the first, is true in any case; consequently the truth
of O follows certainly from the untruth of A. If we
further deny (9, ' some S are not PJ this means, according
to what we said above, * there is no such thing as some
S which are not P,' and this is equivalent to A, 'all
S are P. 9 If we deny E^ ' no is P,' either all or some S,
in any case the latter, are P, and consequently / is true,
'some S are P' : if we deny /, this means, 'there is no
such thing as some S which are P,' and is equivalent to
the affirmation of J, ' no is P.'

78. The two universal judgments of unlike names are
only contrariwise opposed, and we infer the untruth of
the one from the truth of the other, but not the truth
of the one from the untruth of the other. The first case
is obvious: the impossibility of the second follows, after
what we said before, from the consideration that, while
the negation of a universal judgment allows an inference
ad contradictoriam to the truth of the particular of unlike
name, the truth of the latter does not allow an inference
ad subalternantem to that of the universal to which it is
subordinate. Lastly, the relation between the two par-

Chap. II.] PURE AND IMPURE CONVERSION. 109

ticular judgments / and O is called subcontrary opposition.
We infer ad subcontrariam the truth of the one from the
untruth of the other, but not the untruth of the one
from the truth of the other. In fact, the two propositions,
4 some S are not P] and c some *$* are /Y ma Y both be
true together; but if one is denied, the truth of the op-
posite universal follows ad contradictoriam, and from this
again follows ad subatternatam the affirmation of the par-
ticular subordinate to it.

79. I may also mention another logical operation which
has a kindred object. All observations, which always
admit ultimately of being expressed in the form of a
judgment ' $ is /y present us only with that combination
of S and P which actually occurs at the moment of
observation : they tell us nothing as to whether S and P
will be separable or not in other cases, whether, in fact,
there are S which are not P or P which are not S.
Now we have a practical interest in this question which
is very intelligible : we want to know whether a P which
has occurred in S may be considered as a mark> enabling
us to determine the nature of the subject in which it
occurs : in short, whether everything which has the cha-
racteristics of a P is also always an S. The answers to
be expected to this question will accordingly take the
form, ( P is *$"'; and they are therefore called conversions
of the original judgments which gave rise to them. It is
also obvious that we have a special interest in knowing
whether P points to a subject *$* necessarily and always,
or only possibly and sometimes ; whether, as it is ordinarily
put, all P y or only some, are $". Hence it is the quantity
of the original and the converted judgment to which par-
ticular attention is paid, and the conversion is called pure
(conversio pura) when the quantity of the second is that of
the first without any change, and impure (conversio impura)
when it is different, especially when the universal truth
of the original judgment has to be reduced to particular,

no THE THEORY OF THE JUDGMENT. [BookL

in order to make it true when converted. The results
are as follows.

80. The universal affirmative judgment, 'all S are P 9
understands by P either a higher genus in which is
contained along with other species, or a universal mark
in which S partakes along with other subjects. In both
cases there is a part of P left which has nothing to do
with S, and only impure conversion can take place into
the particular judgment ' some P are 6V This rule deserves
attention, for it is one of the commonest mistakes of
carelessness and one of the most favorite means of de-
ception to substitute the universal for the particular infe-
rence, and to assert, ' If P belongs to all S, then S belongs
to all P.' It is true that we do meet with universal
affirmative judgments which admit of this pure conversion,
those viz. in which the extents of S and P exactly cover
each other, and P therefore belongs not only to all S,
but only to all S, so that all P are also S. Such so-called
reciprocal judgments are, 'all men are naturally capable
of language,' 'all equilateral triangles are equiangular';
they can be converted into, 'all that is naturally capable
of language is man,' 'every equiangular triangle is an
equilateral one.' But it is only knowledge of the matter
of fact contained in the judgment in question which can
assure us that the relation, upon which this possibility
depends, holds good between S and P in any particular
instance. Mathematics, therefore, where the pure con-
version of universal affirmative judgments is frequent, are
right in demanding special proof in every case of the
truth of the converted judgment, and by this caution
inculcate the rule that by right of mere logic the universal
affirmative judgment admits only impure conversion into
a particular affirmative. It is otherwise with the universal
negative judgment, 'no S is P. 9 This complete exclusion
of the two concepts from each other clearly holds good
reciprocally, and justifies the assertion that 'no P is S. 9

Chap. II.] CONVERSION OF PAR TICULARS. 1 1 1

The universal negative judgment is therefore convertible
into another universal negative.

81. The particular affirmative proposition, ' some are
/Y obviously yields pure conversion into another particular
affirmative, ' some P are S? And this inference is satis-
factory in all cases in which P is a universal predicate in
which S partakes along with other subjects ; thus the asser-
tion, 'some dogs bite,' is rightly converted into 'some things
that bite are dogs.' But when S is the genus of which P is
a species, as in the proposition, ' some dogs are pugs/ the
only logically admissible conversion, ' some pugs are dogs/
will contrast unfavourably with the actually true one, ' all pugs
are dogs.' The former is no doubt true also, but it ex-
presses only a part of the truth, and in a form which
appears rather to deny than to affirm the other part, that all
other pugs also are dogs. We feel this still more if we start
with the judgment, { all pugs are dogs,' and convert it twice
over. From the first conversion, ' some dogs are pugs,' we
cannot get back again by the second to the original proposi-
tion ; and thus the logical operations have here resulted in
eliminating a part of the truth. This inconvenience could
easily be avoided if the expressions of quantity were re-
garded, as the sense requires that they should be, as
inseparable froni their substantives ; we should then formu-
late the proposition, in the first instance as follows, ' all pugs
are some dogs'; then byconversion, 'some dogs are all
pugs,' and by a second conversion, ' all pugs are some dogs.'
But it is not worth the trouble to improve what are after
all barren formulae.

The particular negative judgment, ' some S are not P,' as
such asserts merely the separability of S from P, not that of
P from S also. The pure conversion therefore, ' some P
are not -5V does not hold good universally, but only of those
P which are predicates common to different subjects, and
are not therefore exclusively dependent upon the nature of
S for their occurrence. For this reason the proposition,,

U2 THE THEORY OF THE JUDGMENT. [Book I.

4 some men are not black,' can be converted into, < some-
thing black is not man'; but the judgments, 'some men are
not pious, 5 * some are not Christians/ would yield ' some-
thing pious is not man/ ' some Christians are not men,' both
inadmissible because piety and Christianity, though not
belonging to all men, belong only to men. These dis-
advantages are in general only avoided by joining the nega-
tion to the predicate, and then converting the proposition,
' some S are non-/Y like a particular affirmative into ' some
non-P are S ' ; e. g. * something not-black, something not-
pious, some non-Christians, are men.'

82. The process necessary in this case has been extended
to all judgments under the name of conversion by contra-
position : in the affirmative judgments the negation of non-
P takes the place of the affirmation of P, in the negative the
affirmation of non-P takes that of the negation of P; the
judgments thus changed are then converted according to
the ordinary rules. In this way we get the following results ;
first, for A, ' all S are P,' ' no 8 is non-P,' and so ' no non-
P is S ' ; for /, on the other hand, * some S are P,' the
transformation into, 'some S are not non-P,' would not,
after what has been said above, allow any conversion, and
contraposition would therefore be impossible ; for , again,
' no is P/ we get ' all S are non-P,' ' some non-P are .'
To carry out these operations in actual instances would pro-
duce unshapely and unnatural forms of expression ; the
substantial meaning of the four forms of judgment may be
given more simply by replacing their quantitative determina-
tions by the equivalent modal ones : even the contraposi-
tion of /, which in itself is impossible, is thus made avail-
able. The conversion of A would then mean, 'If the
predicate P belongs to all individuals of a genus 6", it is
impossible for anything in which this mark is absent to be
an S ' : that of / would mean, ' If P is only known to
belong to some species of S, it is not necessary, but only
possible, that something in which P is absent should not be

Chap, no CONVERSION. \ 1 3

an *': that of E, ' If the mark P is universally absent from,
or contradictory of, the genus *9, it is not necessary, but
only possible, that something which similarly lacks or is
contradicted by P should be a species of S ; ; and the same
inference applies to O also, ' If some 6" are not P, some-
thing which also is not P may be an S but need not
be so/ *

LOGIC, VOL. I

CHAPTER III.

The Theory of Inference and the Systematic Forms.

Preliminary remarks upon the A ristotelian doctrine of
syllogism.

I HAVE pointed out the unsolved problem which compels
us to advance beyond the disjunctive judgment. Before I
follow up this thread of connexion systematically, I think it
will be advantageous to state the theory of syllogism in the
form which it received from Aristotle. I shall not however
follow the original exposition of the great Greek philosopher,
but the more convenient form which came into vogue later.
The writings of Aristotle are preserved, and anyone who
takes an interest in the origin of these doctrines may easily
enjoy his masterly development of them : but when we are
concerned, not with the history of the thing, but with the thing
itself, it would be useless affectation to prefer the inconvenient
phraseology of the inventor to those improvements in detail
which subsequent ages have placed at our disposal.

88. Following Aristotle, we give the name of inference
or syllogism to any combination of two judgments for the
production of a third and valid judgment which is not
merely the sum of the two first. Such production would
be impossible if the contents of the antecedent judgments,
the two premisses, propositiones praemissae^ were entirely
different ; it is only possible if they both contain a common
element M> the middle concept or terminus medius, which

THE FOUR FIGURES. 115

the one relates to 6", the other to P. This medium brings
the two concepts -S and P into connexion, and they can
then meet in the conclusion in a judgment of the form
* S is P,' or, more shortly, S P, from which the middle
concept which served to produce it has again disappeared.
There is no reason in the nature of the case for making a
difference of value between the two premisses SM and
P M but 3, tradition, which cannot be disregarded without
subjecting all established rules to a bewildering change of
meaning, has decided that the premiss which contains along
with M the predicate P of the coming conclusion shall
be called the major premiss, and that which contains S, the
subject, the minor \ the conclusion itself is always conceived .
in the form S P, not in the reverse form PS. This
being presupposed, the further differences in the position
which the three concepts may assume give rise to the
following four arrangements, of which the first three re-
present the three figures of Aristotle, while the fourth
forms that of Galen.

(i)MP (2)PM

SM SM MS MS

SP SP SP SP

84. If we now ask whether, and under what conditions,
these arrangements of premisses, which are in the first
instance merely based upon rules of combination, give
ground for a valid inference, we find at once that S and P
can only be united in the conclusion if the middle concept
remains precisely the same; their union is obviously un-
justifiable as soon as the M connected with *$* in the one
premiss is different from the M connected with P in the
other. Such a division of M would give four concepts in
the premisses, instead of the necessary and sufficient three ;
the avoidance of this quaternio terminorum^ and the securing
of complete identity in the middle term, is therefore the
condition of conclusiveness in all figures alike. To fulfil

I 2

il6 THE THEORY OF INFERENCE. [Book 1.

this condition it is first of all necessary in all figures to
exclude any ambiguity in the meaning of the word which
denotes the middle concept; but besides this there are
special precautions for the same purpose, which the peculiar
structure of the several figures renders necessary, and which
we have now to mention.

85. In the first figure S is included in M in the minor
premiss, M in P in the major, and therefore S in P in
the conclusion. The idea upon which this inference is
based is evidently that of subsumption; that which is a
predicate of the genus is a predicate of every subject of
the genus. This is of itself sufficient to show that the
major premiss in the first figure must be universal ; for it
has to express the rule which is to be applied to the
subject of the minor. The necessity that the middle term
should be identical leads to the same result. For the S
of the minor premiss is always a definite kind or a definite
case of M\ this however is not expressed in the form of
the proposition ; as far as the form goes S might be merely
any kind of M in general ; if this indeterminate M is to
be the same as that which the major premiss asserts to
be P, this can only be secured if the major premiss speaks
universally of all M, thus including the indeterminate cases
along with the rest. It is true that in that case the M
expressed in the major premiss is not identical with the M
of the minor, which, as predicate of , necessarily signifies
only a part of the whole extent of M; but this apparent
difficulty disappears when we consider that the M of the
major premiss which is actually employed in producing the
conclusion is likewise only a part of that which is expressed,
that part, namely, which is intended in the minor. Further,
as the inference in the conclusion depends upon the sub-
ordination of S to -#/", this subordination must be a fact, in
other words, the minor premiss which expresses it must be
affirmative; if it were negative, it would simply deny the
existence of any ground for the validity of the conclusion.

Chap. HI.J THE SECOND FIGURE. 117

On the other hand it does not affect the logical connexion
of the syllogism, but depends merely upon its particular
content, whether the major premiss affirms or denies P of
M y and whether the application furnished by the minor of
the general rule to an instance embraces all S or only some.
The quality of the major premiss and the quantity of the
minor are therefore unlimited. Lastly, the relation, whether
affirmative or negative, in which the major premiss places
M to P, must be transferred unaltered to the unaltered
subject, whether universal or particular, of the minor ; the
conclusion therefore has the quality of the major and the
quantity of the minor. If we suppose all the possibilities
exhausted for which these rules leave room, we get four
valid kinds or moods of the first figure. Their scholastic
names Barbara^ Celarent, Darii, and Ferio, which by the
three vowels in order denote (as every one knows) the
quantity and quality of the premisses and the conclusion,
show at a glance the distinctive feature of the first figure,
namely, its capacity to produce conclusions of every kind.

86. The premisses of the second figure show us two sub-
jects S and P in relation to the predicate M. If both
subjects either have or have not this predicate, i.e. if both
premisses are affirmative or both negative, no inference can
be drawn from them as to a mutual relation between S and
P. For innumerable subjects may all participate in, or all
be excluded from, a mark M, without necessarily having any
other point in common, and in particular without the one,
S, being necessarily a species of the other, P. Only if the
one subject has or has not the mark M always or universally,
while the other is related to M in the opposite way, is there
ground for concluding that the second cannot be a species
of the first. The premisses in the second figure must there-
fore be of opposite qualities, and one of them must be
universal. As however it is the tradition that this second
subject should be supplied by the minor premiss, the pre-
miss in which the first is mentioned, i.e. the major, must be

Il8 THE THEORY OF INFERENCE. [Book I.

the universal one. Thus the conditions of the second figure
may be summed up as follows : the major premiss is uni-
versal, but is not limited as to quality; the minor is of the
opposite quality to the major and is not limited as to quan-
tity; the conclusion is always negative, and has the quantity
of the minor. The possible moods are Camestres, Baroco^
Cesare, Festino.

87. The third figure brings the same subject M into
relation to two predicates, P and S. If M has both predi-
cates, i. e. if both premisses are affirmative, the union of P
and S must be possible, and the conclusion therefore, ac-
cording to the usual logical expression of such a possibility,
is, ' some S are P. 9 The necessary identity of M is in this
case sufficiently secured by the universality of one premiss,
it does not matter which ; for it clearly makes no difference
whether all M have the mark P and only some have 6*, or
whether all Mhzve S and only some P\ in either case there
are always some M which have both and thereby justify the
conclusion, which is always particular, * some S are P.*
Moreover this case, in which M is subject in both pre-
misses, is just one in which its identity might be easily
guaranteed by a word of completely individual meaning,
the proper name of a person for instance. We often meet
with such inferences : in order to prove the compatibility of
two actions which seem to be mutually exclusive, we bring
forward an instance, e.g. * Socrates was P, and Socrates was
also 5,' consequently ' what is may also be P 9 * or * some
S 1 is P.' Logic justifies such inferences by attributing to the
singular judgment, i.e. one whose subject is not an inde-
finite part of a universal concept but a perfectly definite
and unique individual, the syllogistic value "of a universal
judgment. Thus this case comes under the above rule,
Which, where both premisses are affirmative, requires one to
be universal, prescribes a particular affirmative conclusion,
and admits the moods Darapti, Datisi^ and Disamis.

88. Again, if the same subject possesses one of the marks

Chap. III.] TWO NEGATIVE PREMISSES. 119

but not the other, i. e. if one premiss is affirmative, the other
negative, S and P must be separable, or, according to the
ordinary phraseology, the particular negative conclusion
follows, * some *9 are not P.' In this case also it is suffi-
cient for the identity of M that one premiss, it does not
matter which, should be universal, but the minor premiss
must be affirmative. For though one of two marks which
occurs in a given subject is no doubt always separable from
the other which does not occur in that subject, the latter is
not necessarily separable from the former ; it is further
conceivable that if it exist at all it can only do so in
conjunction with the other. Thus life without intelligence
is a possible mark of an animal, but not intelligence
without life. It is therefore the affirmed mark only which
is separable ; only of it as subject can the conclusion
assert that it is not always combined with the other as
predicate; and as this subject of the conclusion is cus-
tomarily furnished by the minor premiss, the minor premiss
must be affirmative and only the major can be negative.
Under this condition mixed premisses yield the moods
Felapton^ Ferison, and Bocardo, these like the preceding
ones having only particular conclusions.

89. Lastly, it is asserted by logic as a universal principle
that in the third as in the other figures two negative pre-
misses yield no valid inference. This is incorrect ; a con-
clusion may be drawn from them similar in kind and equal
in value to those which are derived from affirmative or
mixed premisses. For if the first of these prove that and
P may exist together, and the second that they may exist
apart, two negative premisses prove with equal ground that
S and P are not contradictorily opposed, and that accord-
ingly what is not S need not therefore be P ; in ordinary
phraseology, 'some not-*S" are not P? I cannot see why
this conclusion should stand lower in value than the two
others ; the first only says to us, * when you find S, be pre-
pared for the possibility of finding P,' the second, ' when

I2<J THE THEORY OF INFERENCE. [Book I.

you meet with 6* do not reckon upon the existence of /Y
and similarly the third, * where you do not observe S,
beware of inferring for that very reason the presence of P. 9
In life we often meet with such inferences ; over and over
again, when the necessary presence of some quality has
been over-hastily concluded from the absence of some
other, we appeal to instances in which neither the one nor
the other is found, and so correct an erroneous prejudice by
an inference in the third figure from two negative premisses.
This conclusion therefore is undoubtedly valid, but it would
be an anachronism to invent supplementary names for its
various moods.

90. The premisses of the fourth figure, ascribed to
Claudius Galenus, are in form the counterpart of the first
figure of Aristotle, but do not equal it in value. Its moods
are Bamalip, Cakmes^ Dimatis, Fesapo, Fresiso. As to the
premisses of Bamalip, e. g. ' All roses are plants/ * All
plants need air,' every one who thinks naturally will tacitly
transpose them, and draw the conclusion of Barbara in the
first figure, * All roses need air.' It is true that this conclu-
sion is then of the form PS, but the form SP t which
is required by the fourth figure, can be easily obtained
from it by conversion, 'some things that need air are roses.'
On the other hand .we cannot by conversion recover from
this conclusion in the fourth figure the one which we drew
from the same premisses in the first ; its conversion only
yields the particular proposition, 'some things which are
roses need air/ Thus in this case the conclusion in the
figure of Galen actually loses a part of the truth which
is established by the premisses, a bad recommendation
for a process of inference, the function of which is always to
conclude from what is given as much new truth as possible.
This awkwardness could indeed be avoided, as was shown
before, but the inference would not thereby be made more
natural. Equally unnatural are Calemes and Dimatis^ the
premisses of which will always be transposed by the un-

Chap. III.] REDUCTION. 121

sophisticated mind and applied in Celarent and Darii of
the first figure : they do not indeed occasion a loss of
truth, since the negative conclusion of Calemes admits
pure conversion, while that of Darii is particular like
that of Dimatis. It is only Fesapo and Fresiso which
are less readily reducible to the first figure, owing to the
negative minor premiss which results in both and the
particular major which results in the latter; by pure con-
version of their majors they can be transposed into Felapton
and Ferison of the third figure instead, and this change will
have the same effect of making the conclusions more
natural. In all points, therefore, the fourth figure is a very
superfluous addition to the three figures of Aristotle.

91. Aristotle considered the inferences in all the three
figures to be valid, but only that in the first to be perfect,
because in this figure only does the ground upon which
all inference depends for its possibility, the subordination of
the particular to the universal, find formal expression in the
structure of the premisses. In the other figures too, indeed,
(as he held), the inference rests upon the same principle,
and the relations of subordination, which are necessary and
sufficient for drawing a conclusion according to that prin-
ciple, are contained in the premisses and do not need
supplementing by information from without ; but they are
not exhibited in the actual structure of the premisses ; we
have to look for them there. To make good this formal
defect in the two latter figures, Aristotle has shown us how,
without any change of content, their premisses may be
transformed into those of the first figure. To some people
this has seemed superfluous, and they have objected that
the two other figures also conclude according to principles
of their own and requiring no other evidence : thus the
fundamental idea of the second, that if two things stand
in contrary relations to the same mark the one cannot be
a species of the other, is clear in itself and independent
of the principle of subordination. I doubt this, but shall

THE THEORY OF INFERENCE. [Book I.

not pursue the point further; for to hold that the conclu-
sions of the two latter figures are drawn upon any principle
at all, is to admit that the ground of all inferences is the
subordination of the particular to the universal ; for to what
did those figures apply their principles if not to justify the
conclusion by subordinating the content of the premisses to
them? Aristotle was therefore right in his general idea
of the superiority of the first figure ; we may also share the
interest which he took in justifying the other figures by
these changes of form ; but it is true that in practice it is
seldom of much use to carry them out ; in considering the
fourth figure just now we seemed to find such a case ; the
inferences of the second and third figures are too transparent
to need this assistance.

92. It is therefore sufficient to mention that in the names
of the moods of the two last figures the scholastic logic has
indicated the operations necessary for this purpose by the
letters m s p c. Thus m implies the transposition (meta-
thesis) of the premisses : s and p tell us to convert, purely
(simplidter) or impurely (per accident)^ the proposition whose
characteristic vowel they follow : the meaning of c y reduction
to impossibility (per impossibile duetto), is the only one
which is not quite so simple, and may be at once illustrated
by the case of Baroco. The premisses here are, ' all P are
MJ ' some S are not MJ and the conclusion, ' some S are
not P.' If we suppose this conclusion to be false, it follows
ad contradictoriam that 'all S are P.* If this were so,
and if this new minor premiss, 'all S are P, 9 were sub-
ordinated to the given major, all P are M] it would follow
in Barbara of the first figure that 'all S are M. 9 But this
result contradicts the given minor 'some S are not M 9 } it
was therefore wrong to deny the truth of the conclusion in
BarocO) and that conclusion, * some -S 1 are not P 9 is right.
The other operations scarcely need illustrating. We have
lately seen how, by transposition, #/, of the premisses, and
impure conversion, /, of the conclusion, which was then

Chap. Ill] HYPOTHETICAL INFERENCE. ^23

drawn in the first figure, Bamalip of the fourth is reduced
to the first. Camestres of the second, ' all P are M] ' no S
is M} ' no S is Pj gets by transposition, m, of the premisses
and pure conversion, s y of the minor, the new premisses
* no M is S,' ' all P are M 9 * from which it follows in Celarent
of the first figure, 'no P is '; this conclusion further
requires pure conversion, s, in order to yield ' no S is P,' as
required by Camestres. Darapti of the third figure runs, 'all
J/are PJ 'all J^f are S,' 'some Sare />'; the impure con-
version, /, of the minor gives the premisses 'all Mare P*
'some S are MJ and the resulting conclusion in Darii of
the first figure, 'some 6* are P,' requires no further transforma-
tion, being immediately identical with that of Darapti.

93. Thus far we have conceived of the premisses as
categorical judgments of the form ' S is P.' But the course
of our thoughts may also suggest them in an hypothetical
or disjunctive form. These differences, important as they
are for the judgments as such, are not so for the formal
connexion of the syllogism ; they always belong to its
content, and it is only necessary to take note of them, not
to alter the ordinary syllogistic rules on their account.
This is most obvious where we have two hypothetical
premisses, in each of which two of the three propositions
MSP are connected as protasis and apodosis. Just as
with categorical premisses where MSP denote three
concepts^ the inference in Darii is as follows : ' P is always
true if M is true, M is sometimes true if S is true, there-
fore Pis sometimes true if *S T is true'; in Camestres, 'Mis
always true if P is true, M is never true if S is true, there-
fore Pis never true if S is true'; in Disamis, ' M is some-
times true if P is true, Mis always true if S is true, therefore
P is sometimes true if S is true/

The cases are more peculiar when the major premiss is
hypothetical and connects universally a consequence F,
expressed in the apodosis, with a condition G, contained in
the protasis, while the minor is categorical and affirms or

I2| THE THEORY OF INFERENCE. [Book I.

denies either G or F of all or some instances of S. The
simplest way is to class these cases with the immediate
inferences from judgments, for condition and consequence
are related as subalternans to subalternata. Firstly, then,
the fact that the condition G* is not true in certain cases of
5 does not justify us in inferring ad subalternatam that the
consequence F is not true in the same cases, for the same
consequence may arise from other and equivalent conditions.
But if the condition is true, we infer the truth of the con-
sequence. This gives rise to two syllogisms, since G may
imply either that F is true or that it is not true; (i) ' If G
is true F is always true, G is true in all or some cases of S,
therefore F is true in all or some cases of '; this is a
modus ponendo ponens^ which posits the consequence by
positing the condition, and it evidently answers to the
moods Barbara and Darii in the first figure : (2) * If G is
true F is never true, G is true in all or some cases of S,
therefore ./MS not true in all or some cases of S'; a modus
ponendo tollens, in so far as it does away with the consequence
Fby positing the condition of its opposite, and obviously a
counterpart of Celarent and Ferio in the first figure.

In the opposite direction, ad subalternantem, the truth of
the proposition F in certain cases of S does not prove the
truth of the particular condition G on which it was found to
depend in other cases, for the same consequence F may
arise from several equivalent conditions. But the fact that
F is not true in certain cases of S does prove that all
conditions upon which it could depend, and therefore the
particular condition G, are not true. The following syllo-
gisms are therefore admissible : (3) * If G is true F is always
true, F is not true in all or some cases of S, therefore in all
or some cases of S G is not true/ a modus tollendo tollens^
which by doing away with the consequence does away with
the condition which, had it been true, would inevitably have
given rise to it; it corresponds clearly to Camestres and
Baroco of the second figure : (4) 'If G is true Fis never

Chap. III.] CAUSE AND CONDITION. 125

true, F is true in all or some cases of 6", therefore in all or
some cases of G is not true,' a modus ponendo tollens^
which by positing a consequence denies the condition
under which it would have been impossible ; it repeats
Cesare and Festino of the second figure. Lastly, we may
reflect that the fact that G is not true may also imply that
F is or is not true, in which case we get the syllogisms,
(5) ' If G is not true F also is not ever true, in all or some
cases of G is not true, therefore in the same cases Fis not
true/ a modus tollendo tollens without any peculiarity, merely
translating the ponendo ponens into the negative : (6) ' If G
is not true F is always true, in all or some cases of 6* F is
not true, therefore in these cases G is true,' a modus tollendo
ponens, which was wanted to complete the possible com-
binations of condition and consequence, positive and
negative ; it posits the truth of a condition by doing away
with the consequence which would necessarily follow if it
were not true. An easy change in the form of expression
shows that these two last cases also belong to the second
figure ; the latter of them might be put thus, ( If non-G
is true F is always true, F is always or sometimes not true,
therefore non-G is always or sometimes not true.' As this
exhausts everything that can be proved from the relation of
subalternation, there are no consequences of this kind which
could be classed under the third figure.

94. These syllogistic devices are in my mind of less im-
portance than a circumstance which I never find thoroughly
considered in connexion with the present subject, the
circumstance that all these inferences refer merely to a
relation between the condition G and its consequence F, not
to that of a cause G to its effect F. It is only in the world
of thought that a condition G, if it is once suppose4 to be
true, always has the consequence which by a necessity of
thought belongs to it ; in the real world the cause G, even
if it exists and is operative, may always have its effect F
frustrated by an opposing force U. It being transferred to

12 6 THE THEORY OF INFERENCE. (Book I.

actual events, therefore, all these inferences require to be
modified in ways which applied logic will show us : thus it
is not allowable to conclude that wherever the cause G
operates its effect F is necessarily a fact, nor to assert that,
if is a cause of hindrance to F> where this hindrance
exists F cannot exist ; G also in its turn may be hindered
by a 7, or F may be realised in spite of it by a third cause F.
In pure logic, therefore, it is quite an improper description
of the cases which we have been dealing with to say, that
their minor premiss expresses the real existence of G or F;
the truth is that these two simple letters stand here for
judgments of the form 'S is P'\ it is only the logical
admissibility or necessity of this connexion of thought
between 6" and P which the minor premiss asserts in regard
to certain cases of S, while the major connects it with
another similar relation between S and <2, so as to form an
hypothetical judgment of universal validity. I will not
pursue this point further here ; I have made my exposition
somewhat prolix in expression with the view of indicating
how the matter really stands.

05. If it is true of a subject Z that it is either P, Q, or R^
or that it is both P, Q, and JR, or that it is neither P, Q, nor
J, we first substitute for this triple predicate the simple U,
and call 27 in the first case disjunctive, in the second positive,
in the third negative. If anyone takes the not absolutely
necessary trouble to follow the application of such disjunc-
tive, copulative, and remotive premisses in the syllogism,
he will find these results, (i) If the major premiss is Z U,
and in the minor S Z an S is subordinated to Z, the
ordinary conclusions S U of the first figure follow, and U
has in them the same meaning as in the major : (2) If the
universal major is Z 7, the minor c7, and U is in one of
them positive or disjunctive, in the other negative, we get
the negative conclusions S Z of the second figure with the
quantity of the minor : (3) from the major U Z with a
positive or negative 7, and the minor US with a 7 of the

Chap. HI.] CHAINS OF INFERENCE. 12?

same or the opposite quality, there result the conclusions
Z, always particular, of the third figure : (4) in the two
latter cases, where U having become the middle term
disappears from the conclusion, its multiplicity is entirely
without significance ; what follows follows all the same if the
position of one only of its members in the two premisses be
taken into account. The result is equally little affected if the
universal major Z U has a minor which affirms or denies of
the particular subject Z one of the members of U. If the
major distinguishes only two alternatives and says, * all Z
are either P or <2,' and the minor ' this Z is P ' or ' this Z
is not P, 9 it follows that ' this Z is not Q ' or this Z is Q:
These consequences explain themselves from the nature of
contradictory opposition ; they can be reduced, but without
any conceivable advantage, to the first figure ; * every Z
which is not P is <2, this Z is a Z which is not P, therefore
this Z is a Q. 9 The same unfruitful reflexions may be
extended to a U of more than one member in the major
premiss, for we can always make any number that we
choose of its members into the subject, and say (with only
a bipartite 7), ' every Z which is not P and is not Q is
either J? or T.' Lastly, polylemmas (dilemmas, trilemmas)
are syllogisms with a disjunctive U of many members in the
major Z 7, and the same number of minors, which taken
together affirm of each one of the members of U the same
further consequence T. These are not cases of new logical
forms but only new applications of old ones, and we may
return to them in our applied logic.

96. On the other hand, I have no intention whatever of
coming back to the doctrine of chains of inference. Every
conclusion of a syllogism may conceivably become the
major premiss of another syllogism : the first is then called
the prosyllogism of the second, and each one that follows
the episyllogism of the one which preceded it. A mere
comparison of the names of the moods shows us at once
many properties of the chain thus produced. If its last

128 THE THEORY OF INFERENCE. [Book I.

member is to be universal, the whole series of prosyllogisms,
and therefore the whole chain, must be in the first two
figures ; the entrance of any member in the third figure
produces a particular conclusion, which never leads back
again to universal conclusions. If one of the syllogisms has
a negative conclusion, the conclusions of all episyllogisms
are negative ; and a chain can only end with a conclusion
at once positive and universal if it is in Barbara through its
whole course. It is moreover usual to require, on the
analogy of the simple syllogism, that the major premiss of
the first prosyllogism should furnish the predicate P of the
ultimate conclusion, and the minor of the last episyllogism
its subject S: it would only need patience to find the rules
for the formation of such a series, but I cannot see of what
use they would be. If the conclusion of a prosyllogism,
which is also the major 1 premiss of the episyllogism, is not
expressed, the series give rise to the two forms of Sorites.
The Aristotelian form, ' A is B, B is C, C is Z>, therefore A
is DJ includes each concept in the one which follows ; it
thus proceeds from the lower to the higher, and is produced
by suppressing the conclusions, which we could elicit from
each pair of members as follows, ' B is C, A is B, therefore
A is Q and then, ' C is D, A is C, therefore A is D?
The other form, a late discovery of Professor Goklenius of
Marburg (1547-1628) takes the opposite direction ; its
premisses, *B is A, C is B> D is C . . ./ suppress the con-
clusion of the two first members, ' C is A? which as major
premiss to the third gives the conclusion of the chain in the
first figure, ( Z> is A.'

A. Syllogistic Inference. Inference by Subsumption.
Inference by Induction. Inference by Analogy.

97. The logical truths of which the mind had gradually
become conscious in dealing with its ideas were provision-

1 [' Minor'' premiss, in the Aristotelian Sorites. The author's words
only apply to the Goklenian form.]

Chap. Ill,] SUBSUMPTION. 129

ally summed up by the disjunctive judgment as follows :
every S, which is a specific form of J/, possesses as its
predicate a particular modification of each of the universal
predicates of M to the exclusion of the rest. The problem
which remained was to discover the intellectual operations
by which this required specific mark could be determined
for a given S. This problem is not solved by the Aristo-
telian syllogisms ; they confine themselves to placing the
subject of their conclusion in relation merely with the
universal form of the predicate mentioned in the major
premiss ; so that in spite of the manifold development given
to them and their possible varieties by the acuteness of
earlier logicians, they are merely the expression, formally
expanded and completed, of the logical truth already
embodied in the disjunctive judgment. Like the im-
personal judgment, which, by distinguishing subject and
predicate, made formally explicit a division already indi-
cated in the concept, without telling us anything new about
the mutual relation of the members thus produced, the
Aristotelian syllogism in its first and most perfect figure, to
which we mentally refer the others, merely distinguishes in
two separate premisses the universal rule and its particular
application, which were already similarly related in the
disjunctive judgment. Thus the Aristotelian syllogisms,
constructed as they all are on the principle of placing one
concept within the circuit of another without further defin-
ing its position, may be included, under the general name
of inference by subsumption^ and considered as the first and
most elementary form of the new group of intellectual
operations. We will now attempt to show what is the nexj:
step in advance which they compel us to take.

98. As the most graphic illustration of the idea upon
which inference by subsumption is based I choose the
mood Darii 1 , which expressly brings a particular case in
the minor premiss under the universal law contained in the

1 {Sic. According to ordinary rules the example is in Barbara^
LOGIC, VOL. I. K

I3& THE THEORY OF INFERENCE. [Book I.

major. ' All men are mortal,' says this mood, ' and Caius is
a man/ whence it concludes, * Therefore Caius is mortal/
clearly meaning that by this conclusion a truth which was
not established before is now made certain by the truth of
the two premisses and their relation to one another. But
as early as the scepticism of antiquity the objection was
made, that it is not the premisses which guarantee the truth
of the conclusion, but that the conclusion must already hold
good in order that the premisses may do so. Where, in-
deed, would be the truth of the major premiss, 'all men are
mortal,' if it were not already certain that Caius participates
in this property? And where would be the truth of the
minor premiss, * Caius is a man/ if it were still doubtful
whether among the other properties of humanity he had that
of mortality also, which the major itself alleges as a universal
mark of every man ? Instead then of proving the truth of
the conclusion by their own independent truth, the two
premisses themselves are only true on the supposition of its
truth, and this double circle seems at first to make the
syllogism logically quite inoperative.

99. The weight of this objection is not to be got rid of
by denying it : we will follow out its applications in various
cases. If we suppose the major premiss M P to be an
analytical judgment, if, that is, we assume P to be a fixed
mark without which the content of M cannot be cojnpletely
conceived, then certainly the universal validity of the major
is independently established; but then the minor cannot
subordinate an S to M without already attributing to it this
indispensable P, that is, without presupposing the conclusion
in which that attribution ought first to find expression. If
for instance we reckon weight in the concept of body, we
form the majgr premiss, * all bodies have weight/ without
fear of contradiction ; but we cannot go on in the minor to
call air a body without involving the thought that air too is
heavy, which we are not supposed to know until the con-,
elusion. In general terms, the principle of subsumption

Chap. III.] SYNTHETIC MAJOR PREMISS. 131

requires that the subordinated individual should share the
marks of its universal; but, conversely, nothing can be
subordinated to a universal without already having the
marks which the universal prescribes to it.

The case would be different if we supposed the major
premiss M P to be a universal synthetical judgment. Then
the content of M could be fully conceived without involving
the conception of P, though at the same time we should be
certain, on whatever grounds, that P is always combined
with M. The minor premiss would then merely have to
show in *S the marks which make it an M, and then, and
not till then, the conclusion would add the P which belongs
to S in virtue of its subordination to M, but which had not
before been part of the conception. In the practical
employment of subsumptive syllogisms these assumptions
are always made. When we assert, ' all men are mortal/ we
conceive the physiological character of man to be fully
determined by the rest of his known organisation, and
regard mortality as a mark which need not be explicitly
thought of when we mentally characterise him, because it
follows inevitably from the organisation which determines
our conception. And thus in the case of Caius it is enough
to establish in the minor premiss the fact that he has this
essential organisation, in order in the conclusion to ascribe
to him <ts inevitable consequence. This is still more clear
if we conceive the major premiss as hypothetical, and think
of P as not a fixed and permanent but a fluctuating mark of
M y a consequence which follows upon M under a certain
condition x, a mark which under this condition M assumes
or loses, a state into which it falls, or an effect which it
produces. Then we have merely to subordinate S to M in
the minor premiss in order to conclude that S also, if the
same condition x operates, must exhibit the mark P. And
as a matter of fact this is the form to which most of the
effectual applications of the syllogism in science are re-
cjucible ; they almost all show that S, being a species

K 2

1 3 2 f THE TffEOR Y OF INFERENCE. [Book I.

will develop or experience under the condition x the same
general effect P as we know in M. But as before with the
analytical major premiss the question arose, with what right
the minor could be asserted, so here with a synthetical
major the question arises, with what right we can affirm the
universal validity of this major itself. Mortality is to be a
new mark, necessarily accruing to the organisation of man :
but this universality can only subsist on the assumption that
the conclusion is true, and it falls to the ground if some
capricious Caius is found who does not die. It is clear
what the answer to this will be : * of course/ it will be said,
'every universal major premiss is false if there is a single
instance in which it is not confirmed, and there is always
this danger when the universal in question has been formed
only by an unjustifiable generalisation from a number of
observed instances : but where the necessary connexion
of M and P is inherently demonstrable, the very universality
of its truth provides against the contingency of a single
capricious instance which might contradict it. In the
example before us the matter is doubtful : to the ordinary
mind the universal mortality of man is only an assumption
based upon the overwhelming impression of countless in-
stances, to which as yet no contradictory instance has been
found : to the physiologist, as a consequence of the known
human organisation, it is certainly a matter of settled con-
viction, but not one which can be proved with the exactness
he would wish. But in other cases the universal validity of
the synthetical major premiss is guaranteed either by an
immediate perception, or by proofs which reduce a given
matter to such a perception, and in these cases the syllo-
gism suffices for securing a particular piece of new know-
ledge ; for all that this requires is perfectly practicable, viz.
the subordination of an 6 1 to an M, which here really fulfils
the function of a middle term in connecting S with a pre-
viously unconnected PJ
100. I leave it for the present an open question whether,

Chap, in.] INDUCTION. 133

and how far, the immediate perception of the universal
truth of a synthetical judgment is possible ; for so much is
at once clear, that in any case we shall be only very rarely
in a position to rest the content of a universal major premiss
upon this ground ; countless universal judgments are ex-
pressed and used for inferences, without the possibility of
either themselves passing for immediate perceptions or
being reduced to such by any practicable method of proof.
This wide field of intellectual activity cannot be simply set
aside as invalid, nor can it subsist without logical rules of its
validity. These rules we have to look for, and there are two
which we want. For the effective use of the syllogism it is,
firstly, necessary that we should learn to find universal major
premisses, based neither on an immediate certitude nor
upon the antecedent experience of their truth in every
single instance ; it must be possible to assert the universal
mortality of men, both before it is understood as the neces-
sary consequence of certain conditions, and also before we
have tested every individual man to see whether he is
mortal. A second rule is necessitated by the minor pre-
miss. There are many cases in which we are able to
subordinate an *$" to M because we have found in S all the
marks which ^/"prescribes to its several species, but in most
cases this is impracticable ; even in the case of the Caius of
our minor premiss no one will consider it necessary or
possible, that in order to acquire the right to put him in the
genus man we should test all the properties of his organisa-
tion. If then the really fruitful exercise of thought is to be
possible, there must be a method for finding minor pre-
misses which subordinate a given subject to a genus before
it has been shown to possess fully all the marks of that
genus. The two methods which I am here requiring admit
(though this is not of essential importance), of being at-
tached to somewhat modified forms of the second and third
Aristotelian figures.

101, The problem of all inferential processes is naturally

I34 f THE THEORY OF INFERENCE. [Book I.

this, from given data or premisses to develop as much new
truth as possible ; how this is done, is in itself quite imma-
terial ; the method will be determined by the form of the
premisses, and these we have to take as experience, internal
or external, offers them. Now it often happens that the
same predicate occurs or does not occur, not only in two,
but in very many different subjects P, S, TJ F, W, and the
question is, what consequence can be drawn from the pre-
misses, PM 9 SM y TM 9 VM, , . . , which belong in form to
the second figure of Aristotle. It is clear that in their mul-
tiplicity they do not suggest an inference which would
connect together any particular two of their subjects ; so
far as we aim at such an inference, we can only effect it by
confining ourselves with Aristotle to two premisses and
observing the rules of the second figure. But it is equally
open to us to try whether this recurrence of M in such
different subjects tells us anything about the significance of
M itself, which accordingly would not disappear in the con-
clusion. Such an experiment is what the natural mind
infallibly makes when experience furnishes such premisses,
and it is guided in its experiment by the universal principle
which dominates all its activity, the principle of translating
a given coexistence of ideas into a coherence between their
contents. Where we observe the same mark in different
subjects, we are predisposed to think that the agreement is
not a chance one, and that the different subjects have not
therefore stumbled upon the same predicate each through a
special circumstance of its own, but are all radically of one
common essence, of which their possession of the same
mark is the consequence. P 9 6", T 9 V will accordingly be
different, but still co-ordinate as species under a higher
concept 2 ; it is not as different individuals, but only as
species of the genus S, that they bear the common mark M
as a necessary mark of that genus. Our conclusion there-
fore runs as follows, 'all S are M } ; and in this conclusion
S stands for the higher universal to which we subordinate

Chap. III.] METHOD AND IDEAL, 135

the individual subjects, and for the true subject of the M
which before appeared as a common attribute of those
individuals. Such a process of inference is the simplest
case of Induction, and under this name forms our second
member in the group of inferences based upon the sub-
ordination of manifold elements to the unity of a uni-
versal.

102. This process however seems only to solve imperfectly
the problem which was set to it, that of producing universal
major premisses for subsumptive syllogisms. For everybody
agrees in objecting to induction, that if it is complete its
information is certain but not new, while, so long as it is
incomplete, it is new but not certain. If P, S, T, V are all
the species of 2 which exist, and if each already has a
premiss informing us that it is M 9 the conclusion can only
sum up these premisses in a universal judgment, * All 2 are
J/"'; but it cannot even logically be changed into the general
judgment, * Every 2 as such is M'\ on the contrary, it
remains quite uncertain whether the species of 2 merely
participate as a fact in the common M> and each ultimately
for a special reason of its own, or whether the universal
nature of 2 really contains the one and selfsame reason
which makes M a necessity to all its species. If, on the
other hand, besides those subjects which are combined with
M in the premisses, there are other species of 2 of which
those premisses say nothing, then the conclusion is an
unjustified inference ad subalternantem from the truth of a
limited number of instances to the truth of all, an inference
which may have probability in various degrees, but never
reaches certainty.

It appears to me, however, that these observations, right
as they are in themselves, confuse the pure meaning of a
logical form with the difficulties of its effective application,
and that there was the same error in the criticism made
upon the value of the Aristotelian syllogism. The leading
idea of that syllogism, that every individual derives its right

13<> THE THEORY OF INFERENCE. [Book I.

and obligation to the possession of its predicates through
dependence upon its universal, is without doubt logically a
perfectly valid principle, and exhibits in its true light the
internal construction of the content of thought in question.
It does not lose this logical significance because the truth
of the universal includes or, if we prefer it, presupposes its
truth in all particular instances ; on the contrary, the very
meaning of the syllogistic principle is that the two are
inseparable. Whatever therefore may be the way by which
in practice the mind has arrived at the truth of the pre-
misses, when they are once found the first Aristotelian
figure does express by its structure the inner connexion
of the completed content of thought, though it probably
does not at all express the division of intellectual labour
by which we made it our own. Considered in this way the
subsumptive syllogism is the logical ideal> to the form
of which we ought to bring our knowledge, but it is not the
general instrumental method by which we compose that
knowledge out of the material given to us.

I have a similar remark to make about induction; the
logical idea upon which it rests is by no means merely
probable, but certain and irrefragable. It consists in the
conviction, based upon the principle of identity, that every
determinate phenomenon M can depend upon only one
determinate condition, and accordingly that, wher^ under
apparently different circumstances or in different subjects
P, $*, JJ 7 the same M occurs, there must inevitably be in
them some common element 5, which is the true identical
condition of M or the true subject of M. It would be quite
unjustifiable to object, that as a matter of experience the
same consequence M is often produced by different equiva-
lent conditions, and the same predicate M may occur in
extremely different subjects. Such an objection just shows
the confusion, which we condemned above, between the
logical rule and the conditions of its application. If there
are two equivalent conditions for a result M, it is not in

Chap. III.] ANALOGY. 137

virtue of that which makes them different, P or S, but of
that which is the ground of their equivalence, that they are
really conditions of the same result : so long as we cannot
separate this common characteristic in the two, we have not
yet found the true 2 of the conclusion, and have not there-
fore carried out the induction in the way in which it de-
mands to be carried out. Again, if the same M is found
as predicate in a number of extremely different subjects,
and subjects (as is usually the case in practice) the several
sums of whose marks are only partially known, we may
of course make a great mistake if we combine what is
common to the known marks of all of them, and then
assume it to be 2, the true subject of the mark in question
M. I do not deny that in the practice of induction we are
often placed in such unfavourable circumstances ; but all
these difficulties in carrying out the inductive principle do
not alter its universal logical validity when it asserts, that
wherever different conditions have the same result M, or
different subjects the same predicate M, there must be
discoverable one and only one quite determinate 2, forming
the single invariable condition or the single true subject, to
which the predicate or the result M is to be universally and
necessarily ascribed in a conclusion of the form, ' every 2 is
M. 1 We leave it to applied logic to observe the rules by
which we may succeed in discovering this 2.

103. I introduce the third form of this group under the
somewhat arbitrary name of the inference of analogy. In the
third Aristotelian figure, MP, M S, as in the second, the
structure of both premisses being exactly the same, there is
nothing in their position to lead us to distinguish major
from minor, or to limit their number to two. On the con-
trary, experience will often show us a larger number of them,
M P, MS, MT, M If] will show us, in other words, that
a number of different marks does or does not occur in the
same subject. These data cannot be rejected by the mind,
and it employs them to form an inference which is just like

138 THE THEORY OF INFERENCE. [Book!,

the one described above, only in the reverse direction.
Here, as there, it is guided by the assumption that the
different predicates have not united in the same subject
M by a number of unconnected chances, but that they
must be coherent and owe their coexistence to the
presence of one condition \ they belong to M because
M is a FT, and it is this sum of marks which in its com-
pleteness constitutes the nature of IT ; and M 9 being a
species of FT, has a right to unite them all in itself. Thus
from these premisses we form the conclusion, ' M is a IT,'
and have so executed our second task of finding for the
subsumptive syllogism a minor premiss by which a concept
M (there called S) is subordinated to another concept II
(there called M).

104. Yet this task, like the former one, seems to be but
badly executed, for analogy, like induction, is liable to the
charge that, if complete, it tells us nothing new, and if
incomplete, nothing certain. If the premisses already give
M the marks necessary to make it a II, we gain nothing in
knowledge of fact by actually bringing it under this concept;
the change is merely in the form of our apprehension of the
given content. But in most cases the premisses give only
a part of the predicates necessary to IT, and from the
presence of these we conclude without certainty to that
of the rest, by which alone the whole of II is realised in M.
When we have to do with concrete objects, which in their
totality consist of countless marks, in great part unknown
to us, in part difficult to observe, this is always the case :
from a few properties which we actually observe in an
object, we conclude that it is a metal, an animal of a
certain kind, an instrument for a certain purpose. It is
needless to say that numerous mistakes in the employment
of analogy arise from this fact ; but here also the difficulty
of the application does not diminish the value of the logical
principle. That principle asserts, that no rightly conceived
content of thought consists of an unconnected heap of

Chap. III.] TRANSITION TO MATHEMATICS. 139

marks, which we may increase at pleasure by adding no
matter what new elements; what other marks as yet un-
observed can combine with tne observed marks and what
cannot, is already decided, not indeed by one mark, but by
a given combination of several, in which each is determined
by all the rest ; this is why we are able from the incipient
form of M given us by the premisses to infer its further
completion and continuance ; there is always therefore one
and only one IT, which makes legitimate and possible the
union of marks given in AT, and at the same time the
addition of others not given. This ideal of thought, which
in itself is quite true, only requires, like every form of
thought, to be realised in suitable, not unsuitable, matter.
It is not any casual pair of predicates in an M which suffice
for inferring the rest ; many such combinations may belong
to some other concept II 1 or II 2 as well as to II ; in contrast
with such unessential marks we shall require essential ones
in the premisses, a requirement which is always made in
practice, and which it is left to special knowledge of the
matter in question to meet. The most important source of
inexactness, however, is that all the forms of inference
hitherto mentioned give the predicates only a universal
form, without indicating their measure, specific modifica-
tions, and mutual determination. So long as the premisses
only say, c M is heavy,' { M is yellow/ * M is liquefiable,'
etc., we certainly find in such data no decisive ground for
pronouncing M either to be gold or to be sulphur : but
this is just why such premisses are only met with in abstract
logic; in actual practice attention is always given also to
the particular amount, nuance, and combination of the
predicates, and from this incipient characterisation its con-
tinuity with the completed IT is inferred. It is just this
universal practice of the natural mind for which we have
to find a theoretical basis in new logical rules, and these we
must now go on to consider.

140 THE THEORY OF INFERENCE. [Book I.

B. Mathematical inferences. Inference by substitution.
Inference by proportion. Constitutive equation.

105. I will put together once more, and from different
points of view, the motives which impel us to go beyond
the syllogisms and look for new forms of thought, and for
this purpose I will first touch upon the nature of the judg-
ments which the ordinary theory conceives of as members
of the syllogism. In judgments of the form * is PJ as
I have already observed, language expresses the predicate
with a universality with which it does not belong to its real
subject, and logic usually concedes this when it asserts that
not only does the predicate contribute to the determination
of the subject, but the subject also to that of the predicate.
When we say, ' this rose is red,' we do not mean that it has
a general indefinite red, or any casual shade of colour which
happens to be included under the name red ; it is rose-red
only that we always have in our mind, or, more accurately
still, the precise red of this rose. If then we wished to
express our thought exactly, we should have to say, 'this
rose is red with the redness of this rose. 7 In this apparently
quite barren proposition the logical activity would show
itself in the fact, that the perceived property of the rose is
no longer apprehended as an isolated thing, without any
other home in the world ; in regarding it as a kind of red in
general, which occurs elsewhere and holds good indepen-
dently of this instance, the mind, as we said before 1 ,
objectifies its perception; it gives to what is perceived a
definite place in the world, which makes it something on its
own account, and not a merely subjective excitation of the
percipient at the moment. In this lies the logical gain
which always results when the particular content of a
perception is replaced in the judgment by the universal of
which it is an instance. But at the same time of course
there will be a logical loss, if we get no further than the
i [Above, 3.]

Chap. III.] QUANTITY. 1*41

expression of this Universal, and if the other part of the
perception does not get its due by addition of the parti-
cularisation which is necessary to make the universal named
equivalent to the individual intended. This loss is sustained
by all ordinary judgments of the form just mentioned, and
the Aristotelian syllogisms too confine themselves to dealing
with the universal MOT the universal P.

108. In this way they leave unsolved the particular
problem which the disjunctive judgment suggested, and
fail generally to satisfy the practical needs of thought as a
living process. For already in the disjunctive judgment it
was asserted, that it is not the universal predicate of its
genus which belongs to the individual, but a definite
modification of it, /, to the exclusion of the rest. What
this p is, ought to have been made out by the syllogism ;
and it could only have done so by supplying to the major
premiss, which connects the genus with the universal P, a
minor bringing out the peculiarity in virtue of which S is
this particular species of the genus and no other, and must
therefore have for predicate this and no other modification
of P. This has not been done; the minor premiss also
only mentioned generally the subordination of the indivi-
dual to the genus, but not its specific difference from other
species of it; hence the conclusion could only say what
belongs ^.to the individual as a species of its genus, not as
this species. It hardly needs to be further explained that
this falls short of what the actual processes of thinking
demand. If we argue, ' heat expands all bodies, iron is a
body, therefore heat expands iron,' or, * all men are mortal,
Caius is a man, therefore Caius is mortal,' everyone will
feel the barrenness of this procedure, and will reply, ' Un-
doubtedly heat expands all bodies, but each body in a
different degree ; undoubtedly all men die, but the liability
to die in one man is different from that in another; what
we want to know for technical purposes or for administering
a life-insurance company is, how iron expands in distinction

14* THE THEORY OF INFERENCE. [Book I.

from lead, or how the mortality of Cams is to be estimated
in distinction from that of other men.' This then is what
the new forms have to do ; they have to make the individual
felt as a definite species of the universal, and so enable us
to argue from its distinctive difference to its distinctive
predicate.

107. From another point of view we may notice the fact,
that in logic it has been too exclusively the custom to use
categorical judgments as illustrations, and therefore also to
represent the inclusion of one concept within another as the
most frequent and most important of logical operations.
In the living exercise of thought this is by no means the
case ; we are seldom concerned in practice to determine a
mark which belongs to a concept once for all, or in the
circuit of which the concept is to be classed; most fre-
quently we want to know what variable mark P will occur in
a concept 6* if S is subjected to the condition x. Questions
of this kind are being raised at every moment by life,'
science, and art. We must admit that the ordinary syl-
logistic method does not entirely overlook such cases ; but
it is only an imperfect way of dealing with them to make
P the universal result of the coexistence of x with M in a
major premiss, and then to ascribe P to S, again only
universally, by subordinating to M or MX. What good
is it to say, c if a man is offended he gets angry, Caius is a
man, therefore if he is offended he will get angry'? What
we want to know is, how Caius, being the person he is, will
get angry, and consequently how far we may go with him.
The subordination of Caius to the concept of humanity
helps but little to answer this question ; we must look for
the special characteristics which distinguish Caius from
other persons, and must then have the means of calculating
the effect which offence will have upon these characteristics.
This may be briefly expressed thus : our inferences cannot
be derived from extensive relations between the given
concepts, but only from their content ; without making the

Chap. III.] INFERENCE B Y SUBSTITUTION. % 43

unprofitable circuit through the universal genus, we have
to determine directly from the given marks of a subject,
and from the accruing condition ,#, what new marks will
show themselves or what changes will take place in the old
ones.

108. Considered from this point of view, the new forms
which we have to look for group themselves with the
inferences from analogy. For these also concluded from
the presence, absence, and combination of certain marks
in an S the necessary presence, absence, and mode of
attachment of other marks in the same subject. We may
doubt indeed whether such inferences from content to
content, from mark to mark, are possible on merely logical
grounds, and whether the few which really are possible
are not already anticipated by the familiar logical doctrines
of the compatibility of disparate predicates, the incom-
patibility of contraries, and the necessary choice between
contradictories : statements such as, ' where / is there q
must be/ will after all (it may be said) be supplied by
experience alone, with the single exception, with which
we wish to have no more to do here, when q is already
included in the content of / or / in the extent of q. In
itself this doubt is right ; all assertions about the necessary
connexion or incompatibility of two predicates, with the
exception of the cases last mentioned, can never be based
upon any evidence but that of observation ; but it is still
a question whether logic, with the means hitherto at its
disposal, has made even these necessarily presupposed
facts yield all the consequences which they might be made
to yield : that it has not done so, I can show more shortly
by exhibiting the actual forms of inference to which I refer :
in the natural use of the mind they are current and familiar,
and all that is done here is to give them the place which
belongs to them in a system of logic.

109. Let us leave to the major premiss of our new figure
the form, 'all M are P' or M~P \ to the minor however

144 THE THEORY OF INFERENCE. [Book!,

we will give, not the indefinite form, ' S is an M in general/
but the definite one S=sM; that is, S is that species
of M which we get if we conceive the whole structure of
the marks in J/as determined or modified by the influence
of a specific condition s. The conclusion will then have
to be, * S is cr PJ and it will assert that S, so far as it is
this distinctive species of M characterised by s, possesses,
not the universal mark P, but that specific impression
of it, cr P, which the influence of s must produce in the
structure of M. To avoid misunderstanding, it should
be observed that the influence of a condition s upon the
whole structure of M may transform the different marks
of M in extremely different ways; each one of these
transformations is a result of s, and on that account I
have employed the kindred letter cr in cr P : on the other
hand it is not generally right, though it may be so in
particular cases, to make the modification of a mark equi-
valent to the modifying condition ; therefore the conclusion
here could not be indicated by sP. In the form however
which we have given to the conclusion, it would be merely
the indication, not the solution, of a problem. What is
wanted is to give a name to this <r P, and to show how
P is changed by the influence of s upon M. This remains
impracticable so long as we produce M merely in this
simple form of a universal concept provided with a name :
in order to know how s influences M, we must analyse
the content of M into its several parts, and observe in
what manner they combine. Nobody, for instance, will
undertake to judge how the working of a machine will
change under the influence of a force s, so long as he
merely has the machine before his eyes as a simple object
of perception, M, a steam-engine in general ; he must first
get to know the inner structure, the connexion of the
parts, the position of a possible point of action for the
force s, and the reaction of its initial effect upon the parts
contiguous to. that point. Accordingly, it is only by sub-

Chap. Hi.] THE SYMBOLS. I^g

stituting for the condensed expression or concept M the
developed sum of all its constituent parts, with attention
to their mutual determinations, that we can hope to follow
the influence of s, and so determine, firstly, what is the
whole nature of S which =.f J/J and, as a consequence,
what is the modification <r P of the predicate P which
belongs to this S. As a matter of fact, this second part
of the problem is always included in the first ; the specific
modification of a particular predicate for S cannot possibly
be found without first finding the total change produced
in M by s, on which the modification depends; for if
P were part of a different concept JV, the effect on it
of the same condition s would not be the same as when
it is a part of M. For this reason I shall take no more
notice of the inference to <r P, but shall consider the
problem of the new form to be to determine ^ M, and
give it therefore the form,

Major premiss : M=afrx<:x z . . .

Minor premiss : S =sM.

Conclusion : S =s(a bx ex* . . .)

from which, in regard to single predicates, e.g. ^, there
would follow the definite conclusion, ' S is s . bxj instead
of the indefinite one, '-5* is bx?

110. There is always a danger in expressing very different
and yet connected cases by the simplest possible symbols ;
to avoid misunderstanding, therefore, I add the following
observations. By a, &, t, x I wish to be understood,
speaking generally, different marks of a concept Jlf 9 which,
when completely enumerated, constitute the whole of M.
But in each different concept these marks stand in the
most different kinds of relation to one another, and these
relations are not expressed in my formula ; the double
sign + has been employed as a faint indication of their
possible variety. These signs, -f and , do not suffice for
a full expression even in a case where M does not mean a

LOGIC, VOL. I. L

146 THE THEORY OF INFERENCE. [Book I.

conceptual content of qualitatively different marks, but a
mere whole of quantity composed of the commensurable
quantitative parts a, , c, x. The only symbol of a more
exhaustive kind would be that of the mathematical function
in general, which we used before, M~ F (a, b, c, x . .) ;
but this would have the disadvantage of merely calling up
to thought all modes of connexion between the parts, with-
out giving a sensuous illustration of any. The form of the
series a -f bx + ex* is also an arbitrary symbol the x only
indicates a possible difference of value in the marks, one of
which, x, leaves only one other, a, entirely free, while it
accompanies the rest as a determining condition. The s of
the minor premiss and conclusion appears here as a multi-
plying factor; this is similarly intended to represent to
sense, by the simplest and most familiar form in which one
quantity can influence another, the countless different ways
in which any concrete condition may act upon the manifold
content of any given subject. If we express by a letter
placed underneath on the right any kind of change produced
by a condition in any kind of given matter, and represent
jfcfas a function of a, , t, x (i.e. M=<f> (a, b y c, x) ), we
should in general only be able to represent the conclusion
by =& (a n b n c n x s \ not by S=$ (a,, b n c n #.) ; for it is
obvious that the effect of s may not always be merely to
change the single marks, retaining their general connexion
(as expressed by the second formula), but also (as expressed
by the first) to change this connexion itself; in fact, a con-
dition may so transform the whole structure of a concept
that in its new shape it has to be subsumed under a different
concept M 1 or N instead of the previous M. The ad-
mission which I have now to add makes it unnecessary
for me to go further into this point.

111. The advantage which we anticipate from this figure
of syllogism by substitution, the first of this second group,
depends ultimately upon our knowing what the several
parts of the conclusion mean, i.e. what that value a, or bx 9

Chap. III.] VALUE OF SUBSTITUTION. 147

is which arises from the influence of s upon the developed
expression of M. This, however, if it is not to be learnt
simply by experience, can only be arrived at by thought if
all these mutually related parts are pure quantities, and the
relations between them those of mathematical combination
and separation. Thus the effect in use of our figure is con-
fined to the region of mathematics, and primarily to the
relations of pure quantities. Only the peculiar nature of
numbers, each one of which has an expressible relation to
every other, allows us to disclose the hidden content of Jlf y
by substituting its quantitative parts, in such a way that the
condition s can really operate upon it, and that by applying
the various rules of calculation, by cancelling incompatible
and compounding compatible elements, the change which s
necessitates in M can be really carried out and the form of
the new result exhibited. On the other hand, if we replace
commensurable quantitative parts by incommensurably
different marks of a concept, these advantages disappear
again ; the content of M is only imperfectly disclosed by
such a method of substitution ; for we have no rule here, as
we have in the case of numbers, by which to measure the
effect of a condition acting upon these heterogeneous ele-
ments. It is true that even in such cases we apply the
general idea of substitution : if we want to know how a con-
dition s will act upon a thing, of which we have only the
concept M which its natural history supplies, we analyse M
into its marks ; but the calculation of the effect which s will
have on each and all of them, is based merely upon more
or less indefinite analogies, suggested by experience or some
chance feeling of probability.

112. The fact that the use of the syllogism by substitution
is confined to mathematics, cannot hinder us from giving
it a place in the systematic series of forms of thought. For
in the first place we must not forget that calculation in any
case belongs to the logical activities, and that it is only their
practical separation in education which has concealed the

L 2

148* THE THEORY OF INFERENCE. [Book I.

full claim of mathematics to a home in the universal realm
of logic. But it is not only because they are indispensable
to a part of the work of thought, that these forms have their
place here ; even in those cases where their demands can-
not be realised, they are still the ideals of our logical effort.
For if they can be applied directly to none but quantitative
relations, it is true on the other side that wherever we are
quite unable to reduce the object of our investigation to
those relations, our knowledge of it remains defective, and
that no other logical form can then help us to the answer
which a mathematical treatment of the question, if it were
practicable, would give us. It is hardly necessary in our
days to draw attention to the fact, that natural science owes
its existence to mathematics ; in other fields also we have
learnt to prize the important aid of quantitative statistics in
discovering the laws which govern the combinations of
society ; and even in sciences which from the nature of their
objects are farthest removed from mathematics, we often
feel very clearly the need of connecting them with quantita-
tive ideas. Moral philosophy may decide that every crime
is punishable, without needing a mathematical justification
for the assertion ; but every punishment which has really to
be inflicted must have a measure, and this must be regu-
lated by the measure of badness in the criminal will which
has to be punished. If only it were practicable, the penal
law itself would draw conclusions in our figure of syllogism ;
it would break up every crime by substitution into its
several elements, and from s M, i.e. by calculating the
particular values of the single elements of the crime in this
instance, and so the particular value of the whole, it would
deduce <r P, i.e. the kind and amount of punishment which
the particular instance deserves.

113. There are other things however besides pure mathe-
matics, and science has certainly succeeded in establishing
links of connexion, even between incommensurable pheno-
mena or attributes, which allow us to infer from one to

Chap. HI.] PROPORTION. 149

another. For logic on its part the next problem must be,
to look for the forms in which such inference is possible,
and so to supplement the imperfection of the substitutive
syllogism. It would partly seem indeed that science has
only succeeded in thus bridging the incommensurable by
doing away with the incommensurability, and showing
that two facts, a and , which at first appear to our per-
ception entirely different in quality, really depend upon
quantitative differences between commensurable circum-
stances : I may recall how physics has reduced the quali-
tative differences of our sensations of colour, tone, and
heat to merely mathematical differences in commensurable
motions of commensurable elements. If however we look
more closely at these cases, we find the fact to be, not that
our sensations, a and , are reduced to motions, a, and /3,
commensurable with one another and with the sensations,
but merely that the occurrence of a or /3 and its effect upon
us is represented as the condition upon which the sensation
a or b necessarily arises. The perceived colour a remains
just as incommensurable as ever with the vibration of
ether, a, by which its origin is explained ; and if experience
did not teach us that a is the consequence of , we should
have no logical means of divining from a the nature of its
cause a. What therefore science does in these cases is
really to connect incommensurable elements in a way which
allows us to conclude from one to the other. The original
proposition that a and a, b and /3, do thus mutually point
to one another, is due, as I said, to experience ; in deriving
it from facts the laws of thought are doubtless applied, but
there is no special form of thought involved such as could
solve the insoluble problem of making commensurable what
is really incommensurable. But when experience has in-
formed us of the coherence of two such elements, a and a,
then thought concludes that this coherence will be main-
tained even in the event of their both changing, and that
.therefore a definite change of a into a 1 must always be

l go THE THEORY OF INFERENCE, [Book I.

answered by one and only one definite change of a into a 1 .
Again, these changes themselves, a a 1 and a a 1 , are not
directly commensurable, either in kind or amount : if the
number of vibrations of the sound-wave is increased by the
amount 6= a 1 , i$ is true that a definite increase, daa\
in the heard tone depends upon it ; but this change in the
pitch is a process quite different in kind from the increase .
in the number of vibrations, and cannot be compared with
it ; each quantity can still only be measured by a standard
of its own, and their mutual coherence can be expressed
as a fact and nothing more. But the changes in pitch are
commensurable with one another, and so are the changes
in the number of vibrations ; and if we refer these changes
to d and b as their respective units, we may ask, By how
many units m of the kind d does the pitch change, if the
number of vibrations changes by p. units of the kind 8 ?
m and ft then stand in a purely numerical relation. This
relation may be infinitely various; but, as before, I shall
not indicate the possible variety any further in the form
which I give to this inference. I choose for its name and
scheme the simplest form of proportion, E\e=.T\t y which,
though it only illustrates the case in which mi^is a constant
quantity, still sufficiently symbolises the logical idea implied
in the process.

114. I will illustrate that idea once more by a very
elementary example. Two angles E and e are commen-
surable; so are two segments of a circle 7*and /; but an
angle and a segment are incommensurable and cannot be
directly measured by any common standard : so too the
difference of two angles, which again represents an angle,
is incommensurable with the difference of two curves,
which again forms a curve. Nevertheless, if it is once
established that a certain length of curve t belongs to an
angle e at the centre of a circle of a given diameter, and
if we form the angle E by m times e and the corresponding
curve Tby n times /, then the pure numbers m and n are

Chap. III.] A LIMIT OF KNOWLEDGE. 1'$!

commensurable, which tell us how many times the two
intrinsically incommensurable units / and e have to be
multiplied in order to find two corresponding members
in the two series of angles and curves. For the circle
geometry tells us that m n. Given therefore the two
units, e and /, we only require to know a definite number
E of e in order to arrive at the proper value of T by the
proportion E\e~T\t. Expressed as a syllogism, then,
the whole process would answer to the scheme,

Major premiss : E : e = T\ /.

Minor premiss : E = F (e).

Conclusion: T=F(e.t

115. I need hardly point out that upon this inference by
proportion^ in the simple scheme of which I include all
more complex relations between m and n, rests ultimately
the whole possibility of bringing qualitatively different
occurrences into such mutual dependence as allows us to
calculate one from another. It is also scarcely necessary to
observe that we can only expect this figure to be fully
effective, so far as we succeed in reducing the relations
of things to terms of pure quantity : we should justify this
limitation in the same way as we did the similar limitation
of the syllogism by substitution. In a more lax way we are
constantly judging of things, even in ordinary life, on the
ground of inexact proportions, which mostly pass into mere
comparisons : a general likeness is found between the
relation of a to b and that of a to /3, but the equal exponent
of both is not precisely specified, and so the inferences
drawn generally carry little conviction ; e.g. ' If one of these
relations under a certain condition c has a certain result y,
the other will have a generally similar result under the
same condition/

I have only one more remark to add, in repetition of
what I have already said, viz. that the form of proportion

IJ* THE THEORY OF INFERENCE. [BookL

indicates a limit of knowledge. We find in it the inter-
dependence of two members E and T merely expressed as
a fact, and as such utilised for further purposes; on the
other hand, the question remains unasked and unanswered,
in what way, by what means, through what mechanism, so
to say, the one member E sets about bringing the other T
into any sort of dependence upon itself, especially into this
particular sort. Of course there are a great many com-
posite phenomena, in the case of which this question too
can be answered : scientific investigation, as we said, has
reduced many pairs of apparently disparate properties or
occurrences to merely quantitative differences of commen-
surable terms, and we are then able to see how it comes
about that T must be connected with 2 9 and a particular
increase of the one with a particular increase of the other.
But there is a limit to this possibility: the ultimate dis-
coverable laws of phenomena will always be found to
involve determinate relations between disparate elements,
which we can only accept as facts and utilise in the form
of proportion, without being able to show the reason why
the two elements must be proportionals. We refer many
phenomena to the law of gravitation, the intensity of which
is reversely as the square of the distance ; but hitherto at
any rate no attempt has succeeded in showing how the
distance contrives to weaken the force. We show how the
sensible pitch increases with the increasing number of
vibrations, and how our sensations in general, and in fact
all our mental activities, change proportionally to physical
motions in our organs ; and yet after all, tones and vibra-
tions, mental functions and physical motions, remain for
ever intrinsically incommensurable, nor do we ever ex-
perience how the one contrive to compel the others to
corresponding changes. From one disparate thing to
another our thought has no means of transition ; all our
explanation of the connexion of things goes no further back
than to laws which admit of being expressed in the form of

Chap. III.] RELATIONS ARE IN SUBJECTS. 153

proportion; and these laws make no attempt to fuse the
two elements into an undiscoverable third, but leave them
both in their full difference, and merely point out that, in
spite of their mutual impenetrability, they come as a fact
under a common law by which they mutually determine
one another.

116. In the actual application of the inferences from
proportions another defect, hitherto only briefly indicated,
is tacitly supplemented by attending to an idea which
necessarily accompanies them ; this supplementary idea we
have now explicitly to recognise as having a place of its own
in the systematic series of intellectual operations, the last
place in the present group. In the above scheme the pro-
portion between the changes of two marks E and T was
represented as if it always subsisted between the two marks
as such, it being indifferent in what subject they occur.
Now there are, it is true, predicates which upon logical
grounds, on account of their contrary or contradictory
opposition, or because the one in any case includes the
other, must be either present together or absent together in
every subject : but there are no marks whose quantities and
quantitative changes must always stand in the same pro-
portion to one another, whatever be the nature of the subject
in which they are united. On the contrary, it is just this
nature which determines the exponent of their proportion ;
and the same universally expressed marks E and T y which
in one *S* can only coexist in the ratio n : m, are in another
S l only possible in another ratio n l : m\ Heat expands all
bodies, but the ratios of the degree of expansion to an equal
increase of temperature are different in different bodies.
In practice, where we always have to do with individual
subjects, and have these in mind throughout, we do not
need to state this limitation expressly; but logic is bound
to emphasise the fact that only on the assumption of the
limitation can we talk of using proportions. Nothing but
the specific character of a given subject, in obedience

154 THE THEORY OF INFERENCE. [Book I,

to which all its marks mutually determine one another, jus-
tifies us in concluding from a known value of one of
them to the corresponding value of another according to
a proportion which holds good for this subject only. This
merely brings us back to the idea which lay at the root
of analogy ; for it was only on the strength of the co-
herence of all mutually determined marks in a concept,
that we felt justified in inferring from a limited group of
them to the necessary presence or absence of others, as
we might infer from the beginning of a pattern to its
continuance. This tacitly assumed condition must there-
fore be added in order to complete the expression of the
proportional syllogism, and its major premiss ought to
stand thus, ( If is an M, for this S it is always true
that T: t = E : e. y And the problem which logic pre-
sents to us would not be merely to establish this major
premiss through experience, in order then to bring a par-
ticular case under it in the minor, l S is M ' ; it would
rather be to show how a concept M can be found at
all, such that the proportions required between every two
of its marks can be derived from it.

117. The means for the discovery of such an authori-
tative or constitutive concept have already been indicated ;
they lie in the fact that every mark is determined through-
out by every other, though in very various way 4 s. The
effect of this variety will be that, while in certain cases
the presence of a single proportion between any two marks
is sufficient to determine the rest, in others the know-
ledge of certain essential marks is necessary in order to
deduce the unessential from them, but knowledge of the
unessential is not enough to establish with certainty the
whole content of the concept. But I shall be clearer if
I preface these reflexions by an instance of the actual
realisation of our requirement in the shape of a very
familiar and simple mathematical form of thought. Analy-
tical geometry possesses in the equations^ by which it ex-

Chap. III.] CONSTITUTIVE EQUATIONS. ^55

presses the nature of a curve, just that constitutive concept
of its object which we are looking for. A very small
number of related elements, the indeterminate abscissae
and ordinates in their combination with constant quan-
tities, as constituting a primary proportion, contain, implicit
in themselves and derivable from them, all relations which
necessarily subsist between any parts of the -curve. From
the law expressing the proportionality between the changes
of the ordinates and the abscissae every other property
of the curve can be developed, its course, its openness
or closedness, the symmetricalness or unsymmetricalness of
its parts, the uniformity or measure of alteration in its
curvature at every point in it, the direction of its con-
cavity or convexity, the area which it contains between
any given limits. It is in view of these developments (the
further course of which is too simple to need mentioning
here) that we give the name of inference from constitutive
equations to the method in question. The method itself
is not confined to these geometrical problems; but the
other and in some ways much more interesting examples
supplied by other branches of mathematics, especially the
calculation of variations, cannot be so easily represented
with the simplicity requisite to symbolise the form of
thought which we are considering. Natural science also
could fyrnish approximations at any rate to what we are
looking for. Chemistry would possess constitutive equa-
tions for analogously compounded bodies, in which the
different chemical elements take the place of co-ordinates
and constants, if it could succeed in expressing by its
formulae not only the quantitative proportions of the
elements, but also, more exactly than its symbols at present
do, the rule for the grouping of atoms and the general
character of their interaction.

118. Admitting the objection to the whole of this method,
that like the preceding one it is not fully effective except in
mathematics, we rebut in the same way as we did before

156 THE THEORY OF INFERENCE. [Book I.

the censure which it seeks to convey, and only examine
it more in detail with a view of finding new ways to supple-
ment what is still defective in the method. It is true that
the apparent wealth of development from geometrical equa-
tions is, from a logical point of view, more specious than
real. In order to determine the form of the curve we give
one of the co-ordinates x arbitrary values, calculate the
corresponding values of y from the equation, and then con-
nect the extremities of the perpendiculars (y) erected upon
the extremities of the abscissae (x) so as to form a con-
tinuous line ; the curve is therefore only the geometrical
locus in which the countless results of a countlessly repeated
proportion between different values of the co-ordinates
are combined. As for all the new properties which we
proceed to deduce, concavity, uniform or varying curvature,
closedness or openness, falling or rising of the curve to this
or that side, though at first they look like new marks, they
also are really nothing but relations of magnitude and
position between spatial constructions, relations, it is true,
between different elements, but otherwise of the same
nature as those assumed between the co-ordinates. Starting
with a proportion between two marks x and jy, we do
not arrive at really new marks, qualitatively incommensur-
able with the first ; we advance merely from given homo-
geneous relations to new homogeneous relations, and the
derivability of the latter from the former, as well as their
apparent novelty, depends merely upon the nature of space
and upon the rules which geometrical perception has fol-
lowed in reducing spatial relations to the universal laws of
arithmetical quantities. These inferences therefore are far
from meeting our requirement. The case is very different
when we have to deal, not with mere spatial magnitudes,
but with concrete objects,, in which a number of qualitatively
incommensurable marks are united, and in which moreover
science is unable to explain these primarily incommensur-
able elements as merely different combinations of commen-

Chap III.] CLASSIFICATION. 157

surable ones ; in the face of these difficulties, thought will
still have to look for a form which promises, approximately
at any rate, the same advantages as those which mathematics
with its easier problem offers in full.

119. The group of mathematical forms of inference ends
naturally here, with the emphatic recognition of the fact
that the point which does not admit of being dealt with
mathematically, the disparateness of marks, is precisely the
point which we cannot avoid considering. The place of
the equation will be taken externally by the form of defini-
tion, for this combines a number of heterogeneous marks
into a whole, but distinguishes in them a group of essential
from another of unessential ones ; the former are regarded as
containing the law for the combination of the whole, the
latter as dependent on and determined by the former in
accordance with that law. Lastly, this privileged group of
essential marks can only be found by a comparison of the
given concept with those which resemble it, and thus we
are driven to systematic forms of grouping different things,
and, primarily, to classification.

C. The Systematic Forms. Classification. Explanatory
Theory. The Dialectic ideal of Thought.

120. When we began the account * of the formation of
our concepts, we were already at the opening of the road
which we have now to travel. We already recognised the
matter of an idea to be a totality of different marks, united
according to some definite rule that governed their con-
nexion ; we already expected to find such a rule only in a
group of marks possessed in common by different but com-
parable ideas ; and already we noticed by anticipation the
ascending scale of higher and higher concepts which results
if we continue this process of comparing that which admits
of comparison. I say, * by anticipation,' because the sug-
gestion then made has not so far been turned to account

1 [Sections 20-33.]

150 THE THEORY OF INFERENCE. [Book I.

in the later developments of logical activity. Judgments
and syllogisms based on subsumption have only required
us to consider the one relation which obtains between a
concept $" and its proximate higher universal M- y there was
no occasion for following up the relations of M itself to the
higher grades of the series of concepts above it. For our
only object was to make sure that a predicate P, which, for
whatever reason, belonged to an M } must also belong to
every S that falls within that M, and for this purpose the
logical structure of M itself was to a great extent a matter
of indifference. As middle term it bore the name of concept,
but the character of a concept was in no respect essential
to it; any simple mark, any sum of marks, whether com-
bined under a definite rule, or merely brought together
anyhow in thought, was good enough to constitute such a
middle concept. It was only our concluding reflexions,
which I shall not recapitulate here, that drew our attention
to the necessity that the middle term should be a concept as
we understood it at first, if we are to derive from it the
right and obligation of a subject to possess the marks that
it displays ; for it is only when thus understood that the
concept really forms the complete rule under which the
whole content presented by the subject coheres and is
organised.

121. In saying this we are not simply returning to an
earlier standpoint. In considering the most primary and
simplest forms of thought, the logician can as a rule only
elucidate their results by the use of examples which contain
more logical work than he means them to illustrate. For
these examples must be drawn from language ; and language
is not the expression of a thought which has stood still
where it began, but of the developed thought which has
advanced by a multitude of successive steps beyond the
imperfect results of its earliest endeavours, and which now
conceals the recollection of them under the more elaborate
setting which it has now given to its objects. And it may

Chap. III.] CONCEPT AS LAW OF THING. t$g

therefore seem as if our present problem, the formation of
an essential concept, had been solved already in the above-
mentioned passage; but it needed more than the logical
acts which were then under discussion to generate the ideas
which were there employed as instances ; such ideas could
only arise by help of the processes which have now, familiar
as they are, to be considered in their place in our system.
Thought, in that earlier stage, met the countless multiplicity
of composite images presented by perception, on the one
hand with the desire to grasp each individual image as a
whole whose parts are connected under a definite law, on
the other with the consciousness that such a law could only
be discovered by the comparison of many comparable indi-
viduals and the retention of the common element in all.
But such a comparison depended for the value of its results
on one condition, namely, that the attention which executed
it should be directed to a number of objects S, R, T, whose
common element really consisted in the pervading law of
their whole structure, and not to a number of others 7, F,
Wj differing in all respects except the possession in common
of a limited group of marks. Then, in the beginnings of
thought, there was no logical rule for this selective guidance
of the attention ; on the other hand, it was even then most
effectively secured by the psychical mechanism, which makes
those compound ideas reproduce one another predominantly
in memory which are similar in the whole form of their
connexion, and specially commends them to the attention,
to the exclusion of those whose structure is dissimilar and
whose agreement is confined to isolated groups of marks.

122. In the actual course of its development, therefore,
thought is first directed to those universal concepts which
really contain the law for the complete formation of the
individuals for which they are required ; it is not until it has
some special motive in investigation that it frames universals
in which things otherwise unlike are grouped under a fraction
of similar elements. Thus when we were speaking of the

l6<? THE THEORY OF INFERENCE. [Book I.

first formation of concepts, the current instances of subordi-
nation, e.g. of Caius and Titus to the concept of man, or of
the oak and beech to that of plant, seemed to us quite
natural and intelligible ; it was as if the mere direction to
grasp the common element in the individuals was enough to
put us upon the track of these really authoritative concepts
M. And yet the same direction might equally well have
led us to invent for negroes, coal, and black chalk a common
name N y expressing the union of blackness, extension,
divisibility, weight, and resistance : only the tendencies of
the psychical mechanism favoured the first and hindered
the second of these applications of the logical rule.

123. These tendencies, which have hitherto unconsciously
put us on the right way, we have now to translate into
logical activity ; in other words, we have to become con-
scious of the reasons which justify us in setting up a certain
universal M exclusively as the authoritative rule for the
formation of a number of individuals, instead of some other
JVto which we might have been led by comparing the same
individuals upon a different principle. Logic has shown us
that a single form of interdependence between several re-
lated points gives rise to different results ; we saw that the
truth of the particular followed from that of the universal,
but not that of the universal from that of the particular;
and that while we could always infer from a definite reason
to a definite consequence, a given consequence need not
always lead back to only one reason, but might lead to
several equivalent ones. Applying this to the organisation
of a concept, we find in it certain marks a b c the presence
of which has a determining influence upon the presence,
absence, or modification of others, while the presence of
these others, a /3 y, does not necessarily affect the former,
but is equally compatible with different ones, / q r. This
is the ground for the difference already mentioned between
essential marks, a b , and unessential^ a J3 y ; it is only in the
union of the former that we could expect to find the

Chap. III.] ESSENTIAL MARKS. itfl

authoritative concept for the individuals compared, for it is
only this union which determines the other marks and there-
fore includes none but those individuals which are of
kindred structure throughout ; the latter group of marks, on
the contrary, would leave the former undetermined, and
would therefore, if conceived as a universal, comprise a
number of individuals otherwise entirely different.

124. Our problem accordingly would be, to distinguish
the essential marks from the unessential. This is easy so long
as we have to do with objects which we can observe in
different circumstances ; in that case the variable properties,
which come and go as the conditions change, contrast of
themselves with the permanence of what is essential. It is
different when there is no possibility of such observation,
and where, in the absence of varying circumstances, our
object is to separate the essential from the unessential in
permanent and invariable marks of the same concept : we
have then to substitute comparison of different instances for
observation of changes. Suppose a b c d to be the group of
marks in one case of a given concept ; then, if in a second
case of it d is wanting or is replaced by a quite different 6,
it follows, on the assumption that all the parts of the con-
cept cohere, that the remaining marks also experience a
change; I denote the second case by a l b l c l b, to indicate
that the Alteration of d to d does not cause the entire dis-
appearance of any one of the marks in their universal sense,
but only the transition of each from one of its possible
modifications into another, the form of their combination
remaining the same. In this case d does not belong to the
essential marks ; it is the group A B C, including as modifi-
cations a b c and a 1 b 1 c 1 , which regulates the organisation of
the concept. But this first step informs us only that the
marks united in A B C do as a fact remain together ; it
does not show what internal coherence they have; the
value of the several elements of the group may be very
different ; it is possible that only A B or A C or B C

LOGIC, VOL. I. M

lC2 THE THEORY OF INFERENCE. [Book I.

contain the real law for the formation of the whole, while
the third mark is merely a necessary sequel or allowable
addition to the other two. As the mind is not yet in a
position to investigate the actual object with all the ap-
pliances of science, its only method of deciding this doubtful
question is to continue the same process. We must com-
pare ABC also with instances of the form A B T\ if
the difference of the last mark is here too accompanied
by no more than the previous deviation in the others,
and the connexion of the whole remains the same, the
coexistence and relation of A and B will be the dominant
rule for the original a b c d^ or will represent that union of
essential marks which makes the presence of the rest
possible or necessary, or at any rate determines their
amount, connexion, and relation to the whole. If we
conceive this process continued, we find ourselves on the
way to classification. We can now no longer confine our
consideration to the individual if we would determine its
concept ; that can only be done in this first of the systematic
forms, that is, by investigating its nature in its relation
to others, and judging from its position in an ordered
series what degree of formative influence its several marks
exercise upon its whole nature and behaviour. The au-
thoritative principle of its formation will appear to us to
lie in that inner circle of marks which, when \ve ascend
through the next universal to higher and higher degrees
of universality, remains together the longest and unchanged
in its general form; and the only way to conceive com-
pletely the nature of the particular is to think of this
supreme formative principle as being specialised gradually,
in the reverse order to the grades of universality, by new
accretions which come within the influence of its reaction.

125. The desire to get an explanation of the inner
structure of the composite object by this systematic ar-
rangement, lies at the root of all scientific classification,
but is not equally satisfied by every form of it : before

Chap. III.] ARTIFICIAL CLASSIFICATION. 163

going on to consider the only form which will serve our
purposes here, I will therefore briefly mention, as a pre-
liminary, the artificial or combinatory classifications, which
are designed specially to meet the general demand for
clearness and summarisation, or certain particular require-
ments of applied thought. We first by partition break
up the content of a given universal concept M into its
universal marks ABC...) and each of these by disjunction
into its various modifications which cannot coexist in the
same subject, A into a> a 1 a* . . ., B into b 1 & b* . . ., C intp
& c* c 5 . Then, on the principle of the disjunctive judgment,
every species of J^must possess one modification of each of
the universal marks of M to the exclusion of the rest.
If for the sake of simplicity we confine ourselves to two
marks, of which the one, A, falls by disjunction into only
two members, a and b, the other, B, into three, a, /3 and y,
the binary combinations arrived at in the ordinary way,
a a, a /3, a y, b a, b ft, b y, will comprise all conceivable
species of M. Lastly, it makes the collective survey of
them more easy if we place the modifications of the par-
ticular mark which forms the basis of classification before
the other marks, as was done above, or in the form
M= a (a + ft + y) + b (a -f /3 -f y). The simplest instance of
this classification is the arrangement of dictionaries; the
fixed order of the letters in- the alphabet here gives the
basis of division, not only in the first instance, but also
for the numerous subordinate combinations contained under
the head of each letter. The obvious advantage of this
lexicographical classification is, that it gives a survey of
the material, not only embracing all the words of the
language, that is, all members of the object to be divided,
but also making them easy to find, and this first advantage
it shares with all successful attempts at artificial classi-
fication ; but when we go beyond this we find that the
degrees in which they contribute to the real knowledge,
of their objects are very various*

M 2

1 6^ THE THEORY OF INFERENCE. [Book I.

120. We observe firstly that this method of combination
only takes account of the marks of the given concept
in their isolation, not in that mutual interdependence in
which alone they really constitute the concept. Thus it
is true that the sum of the combinations discovered in-
cludes all species of M, but it may also include others
besides them, which would be true species if the concept
were merely the sum of its marks, but are not true because
it implies their union in a certain definite form which
these other species contradict. The concept of a triangle
does not consist in the fact that we think three angles
and three sides, but in the fact that three sides intersect
one another so as completely to bound a plane space and
by this very fact produce the angles. It is this connexion
of the sides and angles which makes equiangular unequi-
lateral and rectangular equilateral triangles impossible : in
a classification by mere combination these would have
found a place along with the equiangular equilateral, the
rectangular isosceles, and other possible kinds. If the
content of J/, as in this instance, is completely known
and can be exactly constructed, these impossible forms
are excluded by our knowledge of the fact, and the only
use of including them in a provisional classification would
be to stimulate attention to the nature of M, and to the
reasons which make the valid kinds possible <and the
invalid impossible. If on the other hand M is a generic
concept derived from experience, the inner organisation
of which can only be represented imperfectly by description,
not exactly by construction, the species which we have
not actually observed but should have been led to infer
by the method of combination, remain doubtful; further
observation may discover them, further knowledge of facts
may show them to be impossible ; the use of assuming
them provisionally may here also be to stimulate advance
in orue of these two directions.

127, If the method of combination, when applied to

Chap. III.] RELATIVE VALUE OP MARKS. 1,65

objects of experience, is liable to the uncertainty whether
its results do not include more than the facts, it is true on
the other side that, as ordinarily practised, it gives no
guarantee that they exhaust the facts. It is beyond the
power of human imagination to anticipate completely all
the modifications to which a mark may be subject ; our
attention will always be confined to those, / W 8 > which we
happen to have observed ; another modification, / m , which
does not come within the circle of our experience, will be
missing in our classification along with all the species in
which it may possibly occur, and this gap will not be filled
up until our experience has grown. This is the ground for
a logical rule, which is valuable when the decision of a
question involves exhaustive knowledge of all the possible
cases of some object Z ; the rule is to go on dividing and
classifying them by simple contradictory opposition. The
sum of all possible cases of Z is always of the nature Q or
of the opposite non-<2 ; the cases of the form Q are always
either R or non-^?, those of non-(? always either S or
non-; so that at whatever point the division is broken off,
all possible cases are included by it. Such a method,
indeed, is only fruitful when we are so happy in our
selection of the first opposites Q or non-Q, or of all the
subordinate opposites in the same grade, S, non-S, R, etc.,
that we can show without much trouble whether or no the
characteristic in question Z is exhibited in each of the
alternative cases.

128. It is moreover evident that in classification by com-
bination there can be no logical rule obliging us to employ
certain marks at the top as bases of division in the principal
groups, and certain others lower down in their subdivisions.
So long as the concept M which is to be divided is con-
sidered merely as a sum of its marks, without regard to
their mutual relations, any one of them has a right to form
the principal division by its modifications, and any other
may be subordinated to it as basis of a subdivision. The

THE THEORY OF INFERENCE. [Bookl.

obvious disadvantages of this uncertainty are avoided in
practice by concomitant reflexion and an estimate of the
different values of the marks, based upon a knowledge of
the facts or a right feeling, often merely upon an instinctive
taste : all that logic can contribute to these precautions
is the general direction not to choose as bases of division
notiones communes, i.e. marks which are known to occur in
the most different objects without exercising any recognisable
influence upon the rest of their nature. The positive
direction answering to this prohibition, viz. how to find the
decisive bases of division, logic leaves entirely to be given
by special knowledge of the matter in question. And as
regards complex concrete objects at any rate, so long as
fundamental divisions were based upon single marks, the
specialist has always been open to the criticism that he
sometimes removes closely related species to different and
often very distant parts of the system, while he brings
others which are totally and strikingly unlike into surprising
proximity. This is quite intelligible when we consider the
different influence which the marks have on the structure
of the whole concept. There is no reason, for instance,
why the mark B, so long as it occurs in the modified
form $, should not conspicuously affect the formation of the
whole, and in that case all the species under the head of b
will remain connected in form; but the same mark may
entirely lose this influence as soon as it enters into the
group of marks in the modified form /3 ; then the species
under the head of /3 follow all the variations due to the now
influential difference of the other elements A C >, and
examples of M^ otherwise most unlike, now find themselves
in the closest proximity. This is what happened to the
Linnsean system, which selected the number of stamens as
the basis of division ; the result of this view was, that in the
cases where the whole organisation of the plant made the
stamens of importance, the related species were brought
together; where this was not the case, they were separated,

Chap. III.] THE IDEAL IN CLASSIFICATION. 1 67

and different species were united. An instructed taste will
partially obviate this evil also, by selecting different bases of
division for different sections of the whole system. Nothing
but an unseasonable logical pedantry could require that a
system which had begun by dividing its whole object-matter
according to the modifications a b c of one mark A, should
go on to arrange all the groups formed by #, ^, or <r, accord-
ing to modifications of one and the same second mark B ;
it may be that the variations of a mark C are exclusively
of importance for the group with #, and those of a fourth
mark D for the group with b^ and the classification which
proceeds upon this view approaches by that means, and by
that means only, to the real essence of the thing. The risk
which such a method runs of not discovering all the species
completely, must be avoided in some other way ; classifica-
tion does not create the complete material, but assumes its
completeness to be guaranteed elsewhere.

129. Classifications would belong entirely to applied
logic if they aimed at nothing more than complete sum-
marisation, such as is required either when we wish to deal
with a subject practically or when we are just beginning to
consider it logically. But they do more than thus merely
prepare the ground; they themselves represent a logical
ideal, which has its necessary place in the systematic series
of the forms of thought; the very fact that a manifold
material has been brought into the connexion of a classified
system, is of itself supposed to tell us something as to the
nature of each and all of its members, and not to be a
mere preliminary to future enquiry. This appears in the
objections which we make to forced classifications ; we not
only require the lines along which we must look, in order
to find a particular species, to be precisely laid down
beforehand in a series of concepts, but we expect the
actual places in which the several species are found to
correspond in position to the affinities of the species them-
selves. For practical purposes any order will serve that is

1 68 THE THEORY OF INFERENCE. [Book I.

handy for the person who is going to use it, but the order
which logic demands must be true to the facts. Now if we
wish to form a complete idea of any composite object, it
does not matter with which of its parts we begin, provided
only that the order in which we add each new part is
adapted to the particular point with which we have chosen
to start : any idea of a given content so arranged forms a
concept of it, sufficient to distinguish it from others and to
show what it is itself. Amongst these various concepts of
the same M there is one distinguished from the rest by
having for its starting-point the law which determines the
order of all the other marks, and this is the one which we
try to find. We have already given the name of c constitu-
tive' to such a privileged concept ; it might also be called,
in opposition to the mere conceptual form in general, the
logical Idea^ of the object, or, in the vernacular, its thought \
for it is thus that our language distinguishes the 'idea' of
plant or organism, as its formative law, from the concept of
it, which merely comprises the sum of the necessary marks
and the form in which they happen to be combined.

130. It will help us to realise what has just been said if
we mention here two incidental notions which always attach
themselves readily to this search for the Idea of an object,
conspicuously in the attempt of naturalists to improve the
artificial classifications of plants and animals by Deference
to their natural affinities. In these cases we are prone to
regard the universal Idea of animal or plant as a living and
operative force, whose unvarying and consistent activity
gives rise to a series of different forms, accordingly as
external conditions determine one or more of its points of
incidence and oblige it to change correspondingly the
whole course of its action. Another way in which we are
equally prone to regard it is as an unvarying end, which
regulates its modes of operation according to the relations
in which it finds itself placed, and in the different forms
1 [ Idee.*]

Chap. III.] THE 'IDEA' AS END. ,169

which it is thereby compelled to assume realises one and
the same purpose in various ways or with various degrees of
completeness. From this point of view the different species
classified together express the result of the interaction
between the universal idea and the particular relations, with
which as universal it has nothing to do. It will be admitted
that these ways of looking at the matter place it before us
in a clear and vivid light, but it will also be objected that
they are both quite foreign to logic. The objection is
unanswerable; our intention however is not to turn the
ideas of active tendency and purpose to account for the
benefit of logic, but to show that even in their proper place
they only have meaning on the assumption of a purely
logical notion, which we will now explain. If it is to be
possible for the same end to be fulfilled under changing
circumstances, it must also be possible to express its con-
tent by a group of ideas, Z, in which these different forms
of fulfilment cohere as possible species, and from which
they necessarily result if each one of the marks of Z and
each of their mutual relations is successively subjected to
all the changes of which, as parts of Z, they are respectively
capable. If, again, an active tendency is to change its
activity under varying conditions and to manifest itself in
new results, the combination of forces in which it consists
must hp expressible by equations, from which all these
new formations necessarily follow as soon as we give the
quantities entering into the equations all the values suc-
cessively which their natures allow. Activity, then, whether
intentional or unintentional, never produces anything but
what is abstractedly possible to thought, and this becomes
necessary to thought as soon as we affirm one of a number
of related points upon which the rest depend. It is this
which we have in view here : we regard the idea for which
we are looking, neither as the intention of a reflective
consciousness striving for fulfilment, nor as an active force
which causes its results, but merely as the conceived or

170, THE THEORY OF INFERENCE. [Book I.

conceivable reason, the consequences of which under certain
conditions are the same in thought as those which must
follow in reality, under the like conditions, from an intelli-
gent purpose or a causative force. Keeping this in mind,
we may tolerate a phraseology which imports into logic the
idea of an end or of a tendency to development : it will
nevertheless be better to avoid these expressions, and not
to use what is found only in the real world as a name
for the mere reason upon which in thought the reality
rests.

131. Another point which logic cannot neglect may be
introduced here as a sequel to these accessory notions. We
are not surprised in a self-realising tendency if, under certain
conditions, it fails in its endeavour ; and we find it intelli-
gible that an end should be attained under different circum-
stances with different degrees of completeness. Thus both
these notions very naturally give rise to the assumption that
different realisations or examples of the formative idea are
of different values, and that they are not merely co-ordinated
in a general way as species under the universal concept of
their idea, but form within this co-ordination an ascending
or descending scale in which each one has its uninter-
changeable place between certain others. The attempts at
natural classification, which endeavour to satisfy our modern
requirements, are dominated throughout by this thought ;
and it remains to show that this familiar tendency to pass
from classification by mere combination to classification in
the form of a developing series, is justified on general
logical grounds, and that this is the place to justify it.

If, as is too often the case at the beginning of logic, we
regard a concept M merely as a sum of marks universally
expressed, there is no sense in rating one of its species
higher than another. Every .S" either contains all the marks
of its universal M y and in that case it is a species of it, or it
does not contain one or other of them, and then it is, not
an imperfect species, but no species at all of M. But living

Chap. III.] THE PERFECTION QF SPECIES.

thought in actual practice is far from acquiescing in this
hard antithesis; it distinguishes species which correspond
or are adequate to their generic concept in various degrees.
The possibility of making this distinction depends primarily
upon quantitative measurements to which the several marks
and their relations are possibly or necessarily accessible.
The structure of generic concepts, incalculably as it varies
in particular instances, agrees in the main in containing a
number of parts or related points, each comprising a group
of simple marks and standing to the others in all sorts of
relations. By * simple marks' here I mean, not only
sensible properties such as red, sweet, hot, but others also
like heavy, extended, irritable, which, though no doubt they
contain the result of previous observations of complex
modes of behaviour, contain it in so simple a shape that
our logical imagination has long accustomed itself to attach
them to their subjects as stable and simple predicates. To
all these elements of the concept quantitative differences
extend. No mark of any one of its parts is conceivable
without a definite degree of its specific kind of intensity,
and the degrees may vary infinitely; the number of the
parts themselves can, like every number, be increased or
diminished, and every part moreover can alter its logical
value by expanding the simplicity which belongs to it as
a member of the genus into a complex organisation of its
own inner nature; and lastly, every relation between the
various constituents of the concept varies in value according
to the value of those constituents, or admits of greater or
less closeness according to some standard of its own. The
joint effect of all these possibilities of variation is to produce
a number of species noticeably different. If we suppose
that when a mark P of the generic concept M assumes the
value /, the influence which it always exercises upon the
other marks is so intensified as entirely to change the form
of the whole content of M^ the resulting species will no
longer be a species of M, but of some other genus N. And

1 7 2r THE THE OR Y OF INFERENCE. [Book I.

those values of P which approach this decisive limit but do
not reach it, will produce forms which still fall under the
genus M y but approximate gradually to the structure which
is characteristic of N. It is upon this that the difference is
based between species which are more and less appropriate
or adequate to their common generic concept ; each species
is in a certain respect more perfect the farther it is from
passing over into another genus, and that is the logically
most perfect whose divergences from all proximate genera
make up the greatest total amount.

132. I believe I am justified in saying that this point of
view belongs entirely to logic, and is independent of the
views which we may form on other and material grounds
as to the value, meaning, and function of anything which
has the law of its existence in a generic concept. I will
therefore illustrate it by examples which are not affected
by these incidental considerations. The equation of the
ellipse, a 7 ' y 1 + b* x z = cP 2 , leaves the two axes a and b to
be chosen at pleasure, and the formula claims that it will
always produce an ellipse whatever values we may assume
for a and , and even therefore if one of them be assumed
to = o. But in that case the curve passes into a straight line,
and the result which this value gives falls accordingly under
the concept N, that of straight line, which is different from
that of the ellipse. But this example shows at tfie same
time, what we did not choose to assert universally above,
that the extreme species of a genus M^ when produced in
this way, not only must belong to a new genus N^ but may
also continue to come under the former genus M* It is
true that the central equation of the ellipse can tell us
nothing about this case when b = o, because it then ceases
to indicate a curve. But there is another expression of the
essential formation of an ellipse which is still valid ; namely
the rule that the sum of the radii vectores, drawn from two
fixed points on the major axis to one and the same point on
the periphery, is constant and equal to the major axis. In

Chap. III.] LIMITING INSTANCES. 473

the present case where the ellipse has shrunk into a straight
line the two extremities of the line are identified with those
two fixed points, the foci of the ellipse, and for every inter-
mediate point c we have the sum of the distances a c -f c b,
that is, the sum of the two radii vectores^ equal to the length
a b of the straight line.

If a heavy rod of the fixed length a b stands with one end
a on a perfectly smooth horizontal surface, and with the
other b leans against a perfectly smooth vertical wall, the
pressure of its weight makes equilibrium impossible and it
falls. An easy calculation shows that the path described
during its fall by any point C in its length is an ellipse. At
the same time it is clear that the end b must slide down the
wall in a straight line perpendicularly, while the point a must
move away upon the smooth surface in a straight line hori-
zontally. As then every point in the line is affected by the
same group of conditions, these rectilinear motions also
must be regarded as specific forms of the elliptical path
required generically by those conditions. They are in fact
the two extreme cases which we get if we make first one
and then the other axis = 6 ; the end of the rod then
moves in a straight line in the other axis. The middle
point of the rod supplies another singular case ; the axes of
its elliptical path are equal, and thus it describes the arc of
a circle* The nature of the problem before us compels us
therefore to conceive the circle as a species of ellipse, and
the central equation which we have mentioned makes it at
once clear how this is possible. This example therefore
shows us that by changes in the quantity of one of their
parts the species of a genus M approach gradually to the
formative law of another genus, and that there may be
limiting instances which are species both of M and of JV,
because they satisfy the requirements of both concepts ; by
merely examining the actual constituents of such a limiting
instance it is impossible to tell by which generic law its
form is, strictly speaking, determined ; in the present state

174 THE THEORY OF INFERENCE. [Book I.

of our knowledge this question is decided upon incidental
grounds of various kinds.

133. On the other hand, these examples leave an am-
biguity which must be removed in regard to the standard
by which we measure the degree of perfection, or, to put it
shortly, the height of each species. Mathematical figures
have no history telling of their life and growth ; being
merely legitimate possibilities of thought without real exist-
ence, they can be produced for our imagination in the most
various ways, and it is in the abstract indifferent, and in any
particular case depends on the nature of the problem in
question, from what point we begin their construction, or
under what generic concept, what universal rule of con-
struction, we bring them. If we look at them, not geome-
trically, but aesthetically, I mean if we attend to the total
impression of the figure as it is, not to the way in which it
came into being, circles and straight lines contrast decidedly
with ellipses. In the impression of the ellipse as we per-
ceive it the inequality of axes is a necessary element ; on
the other hand it is true that the greater this inequality is,
the more does the curve approach the extreme forms which
we wish to exclude, that of the two straight lines which
coincide with one or the other axis. The characteristic
impression of the genus would be best produced by an
ellipse equally removed from the equation a <=cythat of
the circle, and from the equation <z <=#, that of the straight
line. By combining both equations we might define the
condition of this impression by saying that one axis must
be double the other, and this would be tolerably correct ;
only that a thing cannot be mathematically determined
which does not depend simply on mathematical laws. Our
logical imagination is dominated in every direction by simi-
lar tendencies. Nothing is commoner than for a person
who speaks of a quadrangle to mean really a parallelogram,
or often even a square ; and this inexactness in expression
is very natural ; the imagination wants to realise the concept

Chap. Ill,] DESTINATION AND OBLIGATION. 175

in perception, but can only hold one image at a time, and
it therefore chooses the image which is logically most per-
fect ; and it is the fact that the parallelogram, by increasing
inequality either of the sides or of the angles, continually
approximates to the ultimate form of the straight line, in
which all the four sides coalesce. The observation of
natural objects evinces the same tendency; we always re-
gard as the typical and most expressive examples of each
genus those species in which all the marks are at the
highest value which the combination prescribed by the
genus allows, in which therefore no mark is exclusively
prominent and none is reduced to zero, but all combine, as
far as possible equally, to produce the impression of stable
equilibrium in the whole.

134. I will here repeat an observation which I made
before. I am not afraid that anyone will criticise this mode
of estimating the relative height of species on the ground
that it has nothing to do with logic; its defect is rather
that it starts from inadequate logical grounds, and does not
adapt itself sufficiently to the nature of its objects. To put
it shortly ; that the highest perfection of a species depends
upon the equilibrium of its marks as described above, is the
opinion to which we must come on purely logical grounds,
so long as we have no positive knowledge to supply us
with some other standard of measurement based upon the
essential characteristics of the genus in question. It may
lie in the nature of things that a genus M can not maintain
this equilibrium of marks, but is destined by diminishing
one and intensifying another to pass over into another
genus N\ in that case its species will be more perfect
in proportion as they approach more nearly to this point of
transition at which they cease to belong to their own genus.
We find that the most important attempts at natural classifi-
cation are deeply imbued with this idea of a destination to
be attained, which is constantly impelling the several genera
to advance beyond themselves ; I therefore introduce it here

THE THEORY OF INFERENCE. [Book I.

intentionally, in order to notice its significance for logic,
with which in itself it has nothing to do. We have already 1
separated the idea of productive activity from the concept
of tendency, and the idea of purpose from the concept of
end; we must in the same way separate here the idea of
obligation from the concept of destination. Everyone will
see that the effect of this separation is to do away with all
that is characteristic in the meaning of these three concepts ;
but this is just what we are aiming at. It is not the concept
of destination itself which we are importing into logic, but
merely that of the logical relation upon which it is essen-
tially based, and of which it is itself so graphic an illustration
that we can hardly avoid the term as a figurative expression
of the logical truth. A destination, then, which has to
be reached, differs from a final state which merely happens
to be reached by some process of change ; in the former
case the group of marks which characterises the end attained
contains also the authoritative principle upon which the
marks are connected and upon which they change as they
do ; in the latter, the processes which lead to the end may
take various directions, forwards and backwards, to this side
and that. Bearing this in mind, we can no longer doubt as
to the purely logical sense of the word when we speak of
a 'destination' to which the several genera have to approach.
Hitherto we have looked upon the generic concept M as
the ultimate authoritative principle which regulates the
series of its species, and that species therefore as the highest
which exhibits this concept in the most perfect equilibrium
of its marks; now we are reminded by a consideration
originally foreign to logic, that the case may be different,
and that the formation of the series of species in M need
not really depend on anything in the generic type of M
itself, such as could be discovered by merely examining its
own constituent marks ; that, on the contrary, the formation
of this genus is not rightly explained until we compare
1 [Above, 130.]

Chap. III.] STANDARD OF CLASSIFICATION. 1^7

it with another genus N into which it passes, and with
a third L from which it came by a similar transition, and
these again with those which went before and came after
them; not till this comparison has been made do we get
the direction in which the progress towards perfection takes
place within a higher genus Z, of which L M N are species ;
then, in the series of species in any particular genus M^
those species will be the highest which have advanced the
farthest in the direction in which ^/as a whole is develop-
ing towards the most perfect expression of the higher Z
which includes it It remains to show that this line of
thought, to which we were originally led by an extraneous
suggestion, has its necessary place here in the internal
economy of logic.

135. It is scarcely needful, however, to show this. We
have seen that we could only produce the universal concept,
which includes a number of individuals under it, by uniting
their permanent and common marks ; then we saw that
this constant group of marks might contain elements of
very different values, and in order to separate those which
are not only constant but contain the rule to which the rest
must conform on joining them, we had to compare the
universal already found with other universals, and species
with species ; that which still cohered in this wider field of
change we regarded as the true essence of a genus M y
the species of which were to be ranked higher or lower
in proportion as they realised it more or less perfectly.
But this process has no natural ending ; the same questions
continually recur ; the marks which constitute M will them-
selves differ in value, and the only way to distinguish the
essential from the unessential will be again to compare M
with L and 7VJ to form the higher genus Z from the law
which persistently governs the formation of them all, and to
measure the value of ML IV, as well as that of their several
species, by the degree in which they realise this law Z,
instead of by the degree in which each species expresses the

LOGIC, You I. N

I*/8 THE THEORY OF INFERENCE. [Book I.

more special law of its own proximate genus. This progress
might go on to infinity, or to the point at which we suc-
ceeded in finding a highest ideal A, exhibiting the mode of
connexion to which all kinds of existence, real and think-
able, must conform : from this A a classification might be
derived in the form of a development which evolved from
itself the whole content of the universe, and this develop-
ment, if it were possible, would give the only logical security
that every species had a place in the series of cognate
species answering to the degree of essence which it ex-
pressed. Thus the problem of natural classification leads
of itself beyond the isolated treatment of a particular problem
to the systematic organisation of the whole world of thought.
And this tendency has in fact guided the most important
attempts at such a classification. Those who have wished
to exhibit the development of plants or animals in an
ascending scale, or the events of history (for this form
of thought claims to apply to processes also), have always
been obliged to justify their selection of a particular standard
for measuring the increase in value of the several members
of the series ; this justification they have always had ulti-
mately to find in certain general views as to the meaning of
all being and process, views which are either formally ex-
pressed at the very beginning of the enquiry, or make
themselves tacitly felt throughout it as a guiding principle.

136. Natural classification, then (to sum up under the
traditional name the procedure just described), differs from
combinatory or artificial classification in taking account
of the mutual determination of marks which in the latter
received only subordinate attention, while in its result it is
distinguished by its serial form, in which the members are
not merely placed side by side, but follow each other in a
definite order leading from the province comprehended or
dominated by one species into that of another : this order
begins with those members which answer least to the logical
destination of the whole system, and ends with those which

Chap. III.] TYPE AND IDEAL. '179

express in the most complete and pregnant way the fulfil-
ment of that destination. But the simplest case here
supposed, that in which the series has only one direction,
is not necessarily the only one. In the first place it is
conceivable that single marks in each species may vary
without altering the characteristic structure of the species
at all, so far at least as we can see : in that case the
different instances of this species are equal in value, and
the series may thus be increased in breadth by co-ordinated
members without growing in length. It is also possible
that, owing to different or opposite variations in several
marks, a species M may not only pass over into one
proximate species N, but branch out into several, N, O, Q,
with which it has equal affinity and which contribute
equally to carry out the general development ; these will
then become starting-points for new series, which either
continue side by side or subsequently coalesce again
somehow with the central series. Thus the form of natural
classification in general is that of a web or system of series ;
even the culminating point of the system need not be a
strict unity, for the most perfect attainment of the logical
destination is compatible with a variety of precisely equiva-
lent forms.

137. As the occasion suggests it, I will mention two more
concepts in frequent use, which may find a logical explana-
tion here. The new kind of value which each species
acquires in proportion as it approaches the end to which
they are all developing, does not exclude the other kind
which we mentioned earlier, depending on the equilibrium
which it exhibits in the marks of its proximate genus. The
two values subsist side by side, though the one impairs the
other. We feel the conflict between them in our aesthetic
judgment of phenomena. Every species which expresses
its genus in the stable equilibrium of its marks, impresses
us as perfect, relatively or absolutely : such a species forms
the type of the genus, that type which is the indispensable

N 2

1 8o< THE THEOR Y OF INFERENCE. [Book L

though not the sole condition of beauty in the beautiful,
and which gives even to what is abstractedly ugly the
formal right to a subsidiary place of its own in artistic
representation. On the other hand, species in which this
equilibrium is disturbed by approximation to an end higher
than can be attained within the limits of the genus, give us
the ambiguous impression which we call ' interesting,' like
dissonances in music, which do not satisfy us but prepare
us for a higher satisfaction. Ideal as opposed to type would
mean a phenomenon in which the equilibrium of marks
required to make it typical coincides happily with the
highest development in regard to its logical destination ;
logic does not exclude the possibility of such a coincidence,
and art may perhaps find it realised or be able to realise it
in a phenomenon in repose, though more probably only in
some situation of the phenomenon.

138. Lastly, it will be asked, how classification by de-
velopment reaches its required conclusion, the certainty,
namely, that it has really found that supreme law or logical
destination which governs the particular object or the
universe at large. To this we can only answer, that by
way of mere logic it is quite impossible to arrive at such a
certainty. The form of classification by development, like
all logical forms, is itself an ideal, an ideal which is
demanded by thought, but which can only be realised, so
far as it can be realised at all, by the growth of knowledge.
Nor indeed is this an exceptional condition, such as would
lay this first of our systematic forms under a disadvantage.
The judgment also enjoins a connexion of subject and
predicate which thought has to make if it wishes to come
into contact with its object in its own way; the hypothetical
judgment, for instance, tells us, that only by annexing a
condition to the subject S is it possible to ascribe to it a
predicate P which is not already contained in the concept
of -S"; but logic does not tell us what condition x is necessary
in order to secure this particular P for this particular Sj it

Chap. III.] CLASSIFICATION AND CHANGE. *8i

waits for special knowledge to put its injunctions into
practice. The theory of the syllogism also teaches us how
to draw conclusions when the premisses are given, but it
does not give us the premisses, nor does it guarantee their
truth, except so far as they may themselves be conclusions
deducible from other premisses ; these latter then serve as
the material given to thought, and lead back finally to some
truth which is no longer logically deducible. Similarly
all that the theory of natural classification asserts is, that
every group of complex and coherent objects, and therefore
(since everything coheres) the whole realm of the real and
the thinkable, must be regarded as a system of series in
which concept follows concept in a determinate direction ;
but the discovery of the direction itself, and of the supreme
directing principle, it leaves to positive knowledge to make
as best it can.

139. It is not this objection, but a difficulty of another
kind, which obliges us to continue our enquiry. The
difficulty will be most easily understood by reflecting on
the place which classification occupies in our system. As
a certain arrangement of concepts, it answers primarily to
our first main section, the theory of the concept itself;
but we were obliged to pass on from the concept to the
judgment, for we found changes in the content of thought
which could not be apprehended by conception alone ; on
the contrary, the concept presupposed relations between
its marks which it needed the judgment to interpret clearly.
Classification answers moreover to the first form of judg-
ments, the categorical ; as in these the subject simply had,
assumed, or lost its predicates, so here the supreme authori-
tative concept appears by itself as the sole producer of all
its species, as the source from which they emanate. But
the hypothetical judgment met the categorical with the
objection that a single subject S cannot by itself give rise
to any multiplicity ; and, similarly, all theories of emanation
will have to ask themselves the question, what second

l8a THE THEORY OF INFERENCE. [Book I.

condition it is which makes their first principle develope
at all, and whence come the data in reaction against which
it is obliged to expand into these particular forms and no
others. A corresponding advance is called for here ; and
it will prepare the way if we consider it in still closer con-
nexion with the characteristics of classification described
above. We made it an objection to artificial classification
that it may lead to impossible instances, while in classi-
fication by development we gave proportionately more
attention to the mutual determination of marks ; we as-
sumed that a change in one mark reacts upon the rest,
that through this change one concept passes into another,
and that one species answers better than another to its
concept. This clearly implies that in the formation of its
species the concept depends, not only on itself, or, in
figurative language, on its own purpose, but also on
another power which determines what kinds of realisation
of that purpose are possible or impossible, adequate or
inadequate. This power we have to investigate.

140. The problems of thought are not completely solved
until it has developed forms for the apprehension of every-
thing which perception offers to it as an object and stimulus
of its activity. This requirement, that all thinkable matter
should be included, is not satisfied by classifications. Their
natural objects are always those stationary generic forms
with stereotyped marks, which we believe ourselves to have
before us in perception as fixed points for manifold relations,
but which are far from constituting the whole of what we
really perceive. The several genera are not found in reality
arranged in the system in which classification exhibits them ;
as they actually appear they are always realised in number-
less individual instances, separated in time and space, and
subject to continual change both in their own conditions
and in their relations to one another. Even if we admit-
that the nature of each generic concept contains the law
which every instance of it will obey // it occurs under

Chap, ill.] CONDITIONS OF REAL CHANGE. \^

certain circumstances, yet there is no reason in the concept
itself for the hypothetical addition which we make, neither,
that is, for the presence of that instance at the time and
place at which it is present, nor for the occurrence or
non-occurrence of those particular circumstances. Thought,
therefore, does not embrace in the form of classification
all that there is for it to embrace ; and that which appears
here merely as an incidental stimulus to the universal
concept to produce this or that species of itself, must
also be taken account of as an essential part in the or-
ganisation of the thinkable world as a whole.

141. These considerations are not disproved by the fact
that, as we observed before, classification by development
may extend, not only to generic forms of the real and
the thinkable at rest, but also to progressive processes.
For when it is attempted to represent history as a deve-
lopment, the question what it is which makes process
process, the coming of one state into being out of another,
equally escapes the grasp of logic. When they are re-
flecting on the past or forecasting the future, these specu-
lators may picture to themselves certain situations as
temporary states of equilibrium, which they assume to
follow one another on the stream of events in a fixed and
necessary order; but how the transition from one to
another . actually comes about, they cannot tell us. Nor
could they do so even if they undertook the endless task
of dividing the interval between two such states of equi-
librium into an infinite number of stages ; they would be
able to show that the concept of each stage, when it is
reached, is preliminary to the concept of the next, but
they could not show how the reality which this concept
expresses brings the reality expressed by the other in its
train. We must reflect moreover that in the real world
pure concepts do not occur or develop themselves, but
only particular examples of them, each with all its marks
specifically modified in a way which its concept allows

i8f THE THEORY OF INFERENCE. [Book I.

but does not necessitate. Not only therefore does the
process of becoming remain a mystery which classification
cannot explain, but the result of the process results, not
from the concept of the stage preceding it, but from that
particular realisation of the concept of which also classi-
fication takes no account. All the attempts both of ancient
and modern times to derive the world by way of emanation
from an original concept, are subject to the same defect.
If their original concept is really nothing but the pure
thought of a relation which certain elements not yet named
necessarily imply, all that they can derive from it will
be certain forms, likewise universal, in the shape of possi-
bilities, or, as I have no objection to say, necessary re-
quirements, which in the event of being realised must be
realised in a certain way ; but they have no means of
deciding what this way will be, or of showing where the
desired realisation will come from. If on the other hand
their original thought expresses a relation between elements
not unnamed but definitely characterised, and is endowed
itself with the impulse to development which those elements
do not supply, in the shape of an inherent restlessness
which drives it to evolve its consequences, this is only
to admit that the complete form of each new stage of
development does not depend only on the concept of
the preceding stage, but on the special form in wh,jch, as a
fact, but without any reason, that concept had already
realised itself. It is to admit, in other words, that along-
side of their categorical development by emanation of the
concept out of itself, another power is also at work; this
power, which their theory entirely disregards, consists of
a sum of authoritative hypothetical relations, which ordain
that if the marks in a given concept have as a fact a
certain value, and if certain conditions act upon these
marks, the form of the new resulting concept, the new
stage of emanation, is then, but also not till then, com-
pletely determined Lastly, if we compare the theory of

Chap. III.] CONDITIONS OF CLASSIFICATION. 185

emanation with the method of the inferences by sub-
sumption, we may say shortly that what it lacks is the
second premiss, by which alone they produce from the
universal major the comparatively more special conclusion.
These subsidiary ideas, which are here only tacitly pre-
supposed, logic has to supply explicitly : it cannot stop at
a classification based upon concepts, but must point out
also the legitimate connexion of the judgments which ex-
press the power of a mark already in existence to determine
another which is to come into existence out of it.

142. But it is not necessary to confine ourselves to that
side of classification where it fails to give a complete solu-
tion of the problem of thought ; the attainment of its own
more limited end implies the same tacit assumptions. Each
of the generic concepts classified is necessarily composed
of marks which occur in other concepts as well. It would
be lost labour to construct a scale of genera L M IV, if L
had marks which were heard of nowhere else in the world,
and M and N were distinguished by similar uniqueness.
The marks must rather be looked upon as building-stones
tying about ready for use ; they have to be cut differently
according to their different positions, but they are all of
commensurable material, and it is only the different ways of
using it which give rise to concepts of different structure.
Now in ^classification by development the marks united in
the same generic concept M are spoken of as mutually
determining each other ; a change in one is followed by
changes in another ; and the progress of these changes not
only produces the several species of the genus 3/ 5 but leads
beyond them into the genus N. What rules can this
influence of one mark on another follow but such as involve
a universally valid relation between the natures of these
marks"} And as the marks themselves hold good beyond
the limits of the particular concept M, this relation also
must be independent of M. The formation, therefore, of
the several species of M, their possibility or impossibility,

i 86, THE THEORY OF INFERENCE. [Book I.

and ultimately the possibility or impossibility of M itself, all
entirely depend on what is allowed or not allowed by these
universal taws of connexion between the marks. Accord-
ingly, the classification of concepts cannot fulfil even its
own proper function without presupposing a system of
judgments or universal laws regulating the admissibility,
mode of connexion, and mutual determination of all marks
which are to be united in this or that generic concept.

143. I must mention here an apparent contradiction, the
removal of which will conclude these preliminary considera-
tions. We have already, in treating of the form of propor-
tion, spoken of the necessity of this mutual interdependence
of marks ; we there corrected ourselves by saying, that when
a constant relation exists between two marks, the measure
of their interaction is not found in the marks as such, but in
the nature of the whole in which they occur or in the con-
cept of that whole. We seem here to be retracting this
statement, but we are in fact confirming it. For the very
point which we have now made clear is, that the content of
the concept, to which we there transferred the decisive in-
fluence, is nothing but a number of marks, each extending
beyond the concept itself, and all connected in it in a
definite way. Between these marks, as we saw, different
relations are possible ; it may happen that the idea of one
involves that of another ; in that case every subjeqt which
has the first will have the second also ; or it may be that
two marks exclude each other as contrary and contradictory
members of a common element, and in that case there is no
conceivable subject in which they can exist together ;
between these extreme cases lie others, in which, without
any similar logical grounds, we perceive two marks to be
combined as a fact, but the value of the one does not
always imply a like value in the other. These are the cases
to which our observation above applied; for the reason
which narrows the range of this variation, and fixes the
precise proportion in which two marks determine each other

Chap, III.] EXPLANATORY THEORY. 187

m any particular object, lies in the simultaneous presence
of all the other marks, in the values and the mode of their
combination. What was undecided in the relation of the
two is decided by their relations to the rest ; if the different
equations, by which we may suppose the latter relations to
be expressed, are only satisfied by one value of each of the
marks, the formation of the whole is completely defined ;
where the number of equations is not enough for this, the
whole is still partially indefinite, and exhibits a universal
concept in which there is still a possibility of different
species. Thus it is true that the concept determines for its
subordinate species the proportion in which each pair of
marks condition one another ; but it only does this in virtue
of the ordered sum of its other marks, and so far as these are
known to have definite values. Our method, in fact, has
always been based upon this supposition. In proposing to
classify a generic concept by developing its species out of
it, we have always had to assume that certain of its universal
marks are already defined by their places in the series ; not
till then could the rest acquire that definite character which
was necessary to complete the distinction of one species
from another. In the concept itself the existence of this
primary definiteness, of which the rest was a consequence,
was only a possibility ; its realisation was assumed in
thought .independently of the concept.

144. If we sum up these considerations, we may say that
every individual and every species of a genus is what it is
through the co-operation of the complete sum of its con-
ditions ; these conditions consist in the fact that a number
of elements or marks, which might also exist in separation,
are as a fact given in a certain combination, which might
conceivably be different, and each with a certain quantita-
tive value, which is one amongst other possible values.
From this given union of conditions, according to universal
laws which hold good beyond the limits of these elements,
this perfectly definite result follows. Every such result,

1 8$ THE THEORY OF INFERENCE. [Book I.

when it is once there, can be compared with others, and co-
ordinated with them as species with species or subordinated
to them as species to genus; but these concepts, which
hitherto we are considering as the key to the understanding
of the structure of their subordinates, must not be credited
with any mysterious and authoritative power, beyond the
fact that they are condensed expressions for a definite union
of separable elements, which act and react upon each other
according to constant and universal laws, and give rise in
one combination to one set of results, in another to another.
145. It is evident what a revolution these considerations
cause in the whole view of logic : we see it in the logical
form of explanatory theory which modern science opposes
to that of classification, by which antiquity was exclusively
dominated. I leave it to applied logic to speak of the
methods which this change in our thoughts necessitates in
practice, and confine myself to pointing out briefly how the
logical view of the world, if it were attained as these theories
understand it, would differ from that of the theory of classifi-
cation. In the first place, we hear no more of a categorical
emanation of all real and thinkable matter, proceeding by
the mere impulse of a plan of development contained in the
point from which it starts, without the aid of any other
conditions ; the form of science becomes essentially
hypothetical. It does not describe what is and wha>: comes
to be; it defines what must be and come to be //"certain
conditions are given ; the question whether, and in what
order and connexion, these conditions occur, is excluded
from the province of logic and left to be answered by
experience, which will bring the facts to illustrate the
application of the theory. Nor will I here raise the question,
how this theory gets at those universal laws by which it
decides, that wherever a particular group of conditions is
given, one particular result and no other must occur ; it is
sufficient at present to observe that it does start with this
conception of a law which fixes the particular result of a

Chap. III.] < MECHANICAL' SCIENCE. 1^89

particular condition universally. This means, that wherever
the condition a 4- b is found, only c follows from it, and the
nature of the object in which a -f- b is found has no power
to give this condition directly any other result than c\ it
can only do so when other conditions, a -j- d, are present in
it as well as a + , and the former co-operating with the
latter oblige c to change into y ; and this co-operation also
takes place by a universal necessity quite independent of
the nature of the particular object and equally binding upon
all others. And in the new result y the law which connected
c with a -j- b is not eliminated, but continues to operate
concomitantly; for a + d alone would not have produced y,
but 3.

From these universal laws arises that mechanical character,
of which the adherents of these theories make a boast, and
their logical antagonists a reproach. The tendency to derive
a series of phenomena ' organically,' as the phrase is, from
the meaning of a conception which develops itself in them,
is met by the assertion that a mere meaning which wants to
develop itself does not produce anything, but that everything
exists, and exists only, when the complete sum of conditions
is given from which it follows necessarily by universal laws ;
it must be regarded as the result of these conditions alone,
and explanation consists merely in showing that a given and
perfectly determinate thing is the inevitable consequence of
the application of universal laws to given and equally de-
terminate circumstances. Animated by this logical spirit,
which is found most pronounced in the mechanical sciences,
explanatory theories are averse both to using and looking for
universal generic concepts, and to schemes of classification.
According to them a phenomenon has been merely ob-
served, not understood, as long as it can be referred only
to the special characteristics which distinguish one concept
from others, and not to the prescription of a universal
authority which is equally binding upon everything thinkable
and everything real. It is their pride not to need generic

190 THE THEORY OF INFERENCE. [Book I.

concepts and their arrangement in a system of classes, but
to show that, whatever the context from which a phenome-
non gets its meaning, we know all about it as soon as we
know the sum of related points combined in it ; for whatever
is, is merely an example of what must come to be when the
universal laws are applied to this or that particular group
of given elements. Even the position which is sometimes
taken up as the utmost that can be conceded on the other
side, does not satisfy the demands of these theories, the
position that everything obeys universal laws, but each
domain of reality its own, and that the laws of living and
spiritual existences are different from those of lifeless and
material ones. It is indeed obvious that those special laws
to which any given phenomena are immediately subordinate,
and with which therefore they are most closely connected
in matter and form, vary with the varieties of the subjects
which they express ; but there could not be two worlds
depending on two supreme and independent laws, unless
they had nothing to do with each other and no effects from
the one were ever felt within the limits of the other : anyone
who speaks of one world, embracing those different groups
of self-developing things and events, must start with a single
law valid for all reality, or a single unbroken circle of law,
of which all the special laws of different domains are par-
ticular cases, and from which they arise as sooft as it is
supplied, in a succession of minor premisses, with the
different conditions which differentiate the several domains
of active existence.

148. In accordance with my plan of dividing the problems
of logic, I have omitted from the preceding account of ex-
planation all mention of the means which the theory employs,
partly for discovering the universal laws which it assumes
each coherent group of existence to obey, partly for de-
tecting in the manifold variety of experience those inner
coherences themselves which the subordination of different
elements to the same common principles admits or requires.

Chap, III.] AESTHETIC OBJECTION. 19!

I have reserved to applied logic the utmost freedom to
follow the course of these efforts ; all that came within our
systematic survey of the operations of thought, of which we
are now approaching the conclusion, was the form which
explanation would like to give to the connexion of all
thinkable matter, and in which, if it could really be given
completely, the final goal of intellectual aspiration would
seem to be attained. As to this goal itself, however, I do
not share the prevailing conviction of the present day.
Explanatory theory is almost the only form in which the
scientific activity of our time exhibits itself; the conscious-
ness (so late in making itself felt) of the principle which
that theory has to follow, strongly separates all modern
science from that of antiquity and the middle ages, and the
methods of investigation developed in consequence of it
form the precious treasure which places the modern art of
discovery far above that of ancient philosophy. Yet the
opposition so unremittingly made to this form of thought,
when it claims exclusive dominion over the thinkable world,
shows that the belief that it leaves nothing more to wish for
is not universal. If we consider first the familiar forms
which that opposition assumes in our collective view of the
world, we shall be able to disengage from it the purely
logical residuum of feeling which the explanatory theories
fail to satisfy.

147. The assertion that all existence is subject only to
universal laws, and that every individual is nothing more
than it must become according to those laws, if conditions,
which might have been combined differently, have as a fact
combined in a certain form, is most obviously distasteful on
aesthetic grounds and to artistic natures. Beauty, it is felt,
cannot be understood upon such a view ; it only seems of
value, and to be really itself, if the ultimate form which
excites our admiration is the result of a single power, a result
which is indeed inevitable, but which, besides being inevit-

ip? THE THEORY OF INFERENCE. [Book I.

able, is also the fulfilment and manifestation of a living
impulse : it would appear unintelligible, if it were merely
a lucky case of harmony between casually coincident
elements. I have tried elsewhere to show that this
aesthetic objection is wrong, if it goes on to deny the
universal validity of the explanatory or mechanical theory.
As understood by that theory, the meeting of the various
conditions is never a matter of chance, but always the
necessary consequence of the past states of the world. If
we follow out this thought, it leads us back to some com-
bination of elements which we regard as the initial state of
the world; and there is then nothing to prevent us from
supposing that this combination, which might conceivably
have been different, contained within it the marvellous germ
of beauty, which, making itself felt through the whole
mechanical chain of consequences, gives birth by single
acts of its own to the beauty of single phenomena. Or
again, if we wish to avoid the difficult conception of an
initial state, there is no reason why we should not take a
section, as it were, of the world's course at any point of
time that we choose, and suppose the combination of all
the forces then acting simultaneously, just because it is
that combination and not any other equally conceivable,
to be the one and sufficient reason of all individual beauties.
Such a supposition would give room for everything which
our aesthetic feeling considers necessary to maintain the
dignity of beauty ; it would merely have somewhat changed
the place of the single impelling power ; this power would
no longer lie self-centred in the individual beautiful thing ;
it would continue to be active in the individual, but only as
the after-effect of a universal which permeates all indivi-
dualities. By thus putting back the origin of beauty we do
not run counter to aesthetic requirements; on the other
hand, the mechanical theory, obliged as it is to assume
some existing state of things in which the continuity of
development according to universal laws is exhibited, has

Chap. III.] REALITY AND LAW. 1,93

no motive for conceiving that state as meaningless rather
than full of meaning, as irrational rather than rational, as
the source of caprice in the world's course rather than of
consistent purpose. There is however one point which the
requirements of aesthetic feeling and the admissions of
scientific explanation equally imply, namely, that the
secondary premisses, which we bring under the universal
laws and by which we denote the facts to which the laws
apply, cannot have the casual origin which they doubtless
seem to us to have when we are absorbed in some particular
field of enquiry and have taken them out of their mutual
connexion. They must themselves be systematised and
form parts of a whole, that whole which comprehends all
real objects to which the universal laws apply. The minor
premisses to our general view of the world must not be
conceptions of a number of disconnected possibilities in
hypothetical form, each of which, if it occurred, would lead
by universal laws to a definite result ; they ought to dis-
tinguish categorically each possibility which occurs from
those which do not occur, and exhibit it as a legitimate
member with a place of its own in the universal order of
reality.

148. This requirement is partly supported, partly modi-
fied, by metaphysical considerations. For what would be
the meaning of assuming on the one side a realm of
universal laws, and on the other a sum of reality which
conforms to them, if no further relation existed between
the two and made this subjection intelligible? And in
what could the subjection consist if not in the fact that the
behaviour prescribed by the laws is from the very first an
actual property of all reality, a constant mark alongside of
the different or changeable marks by which one real thing
is distinguished from another ? No truth at any rate can
be applied^ as we are in the habit of saying, to a given
content, unless the content itself answers to it ; every appli-
cation is merely the recognition that what we wish to apply

LOGIC, VOL. I. O

191 THE THEORY OF INFERENCE. [Book!.

is the very nature of that to which it is to be applied. Now
a limited number of observations enables us to discover
that everything real exhibits certain constant characteristics,
and these characteristics then take the shape in our mind
of expectations which will be confirmed, and which we
bring with us when we make further observations ; thus we
easily come to regard them as something which exists inde-
pendently in fact as well as in our thoughts, and is prior to
the object in which we shall find fresh confirmation of it ;
hence all that strange phraseology which regards universal
laws as powers ruling on their own account, to which every-
thing real, whatever its origin and whatever its nature, is
subsequently obliged to submit. If we avoid this wrong
conception, and connect that which we substitute for it
with that to which our aesthetic requirements give rise, the
one and undivided object in which our thought now seeks
satisfaction is a being, which, not in consequence of a still
higher law but because it is what it is, is the ground both
of the universal laws to which it will always conform, and of
the series of individual realities which will subsequently
appear to us to submit to those laws. I have no intention
of exhausting this subject here, and I pass over many
difficulties which we shall have to notice later, some of them
in the course of our present logical enquiries, others in
their metaphysical context : it is enough here to follow out
the logical form of thought which the mind must look for if
it tries to satisfy the want just described.

149. This form will no longer be quite that of inference
as described above. The universal law, to which the major
premiss there gave the first place, instead of standing out
from the other elements as their essential condition, will
now accompany them as a latent idea, always understood
but not expressed ; its former place is taken by the universal
nature of the sum of existence, which is developing itself in
the world. Nor is this nature conceived as an ideal content
at rest, which could not be set in motion without extraneous

Chap. III.] FORM OF SPECULATIVE THOUGHT. 1^5

conditions, but as the subject of a movement which enters
into its very constitution and without which it would not be
what it is. The particular form which the moving content
assumes at each successive moment, depends on the one
side upon its permanent purport and permanent direction,
on the other upon its particular position or the particular
point to which it has thus far developed, not through ex-
traneous influences- but through its own movement. It
would be possible, but would only lead to prolixity, to
express the essential truth in this kind of idea without im-
porting into it the conception of motion ; we should then
find ourselves requiring an idea which includes in the
system of its species and sub-species the whole of reality ;
but the differences and the order of these species would
not be determined independently of the idea by pre-existing
marks and their modifications ; the idea itself would contain
the reason for the presence of the marks, for their possible
divisions, and for the arrangement of the resulting varieties
according to their value, in fact the whole reason for its
own classification. We may formulate our requirement
most shortly as follows : the form of thought for which we
are looking must have only one major premiss for all its
conclusions, and this premiss must express the movement
of the world as a whole ; its minor premisses must not be
given to jt from elsewhere, but it must produce them from
itself in the form of necessary and exhaustive varieties of
its meaning, and thus must evolve in an infinite series of
conclusions the developed reality which it had conceived
as a principle capable of development in the major premiss.
150. It cannot be said that the impulse to organise the
whole world of thought upon this pattern is foreign to the
mind when left to itself ; it has been at work at all times,
and whenever a view of the world more or less like the
theory of mechanical explanation has developed itself, this
impulse has met it with the reiterated demand that the
world and all things in it should be regarded as a living

O 2

1 9^6 THE THEORY OF INFERENCE. [Book I;

development. For it is in the phenomenon of life that we
believe ourselves to see these claims of the mind com-
pletely satisfied ; as there the original type of the organism
is made into the efficient power which produces the incen-
tives and conditions for its own consistent development, so
we would have the world as a whole evolve from itself the
occasions which are the necessary conditions of its gradual
self-realisation. We need not here notice the errors in this
belief in the independent development of the individual
organism; it is enough that it appears to be a graphic
instance of what we are looking for. The same image has
also been a constant favorite with the theory which, for
the last time in our day, avowedly aspired to a vision of the
universe springing out of the unity of an idea, which
develops itself and creates the conditions of its progress.
For it was in no attitude of investigation and reflexion, by
no means of logical and discursive thinking, bringing in-
dependent minor premisses under universal majors, that
the Hegelian philosophy even wished to derive the world
from its single principle : it only proposed to look on and
see how the development followed from the inherent impulse
of the idea. And for this intellectual vision, this ' specu-
lathe"* thinking in the original sense of the word, it believed
itself to have found a guide in the dialectical method, a
guide which enables the spectator to follow the true course
of the self-realising development. I shall still keep to my
principle of saying nothing in this survey of logical forms
about the practical rules for securing their application to the
matter of thought, and therefore leave for a later occasion
what is to be said about this method as a method ; but I
shall appropriate the antithesis between speculation and
explanatory theory for the purpose of describing the final
shape which we aim at giving to all thinkable matter, and
call the form of speculative thought this third member, with
which the series of comprehensive and systematic forms
comes to an end.

Chap. III.] FORMS OF THOUGHT AS IDEALS. itf

151. And yet I feel that I must not conclude quite so
shortly ; I must return once more to an observation which
I have already made. All forms of thought which we are
considering are ideals ; they indicate the final shapes which
thought wishes to give, or to be able to give, to the matter,
great or small, which it has before it, in order to satisfy its
own inherent impulse by showing the coherence of all that
coexists. Nor is the validity of these ideals at all impaired
by the fact that human knowledge is not able to apply them
to every given instance. It may be that we are not always
in a position to discover the universal laws which govern a
particular circle of phenomena ; and it may be that, if we
had discovered them, we should not succeed in bringing all
particular cases under them so completely that the necessity
of any given result was at once apparent. But we should
not push forward our enquiries in this direction so untiringly,
if we were not convinced that the principle of the explana-
tory theory is universally valid, and that its validity is
independent of our present ability to verify it in every
conceivable instance. Perhaps the form of speculative
thought is in a still more unfavorable position ; the con-
ditions under which human thought is placed may be
altogether inadequate to achieve the speculative ideal in
more than a few instances, perhaps even in one ; yet this
ideal also- will retain its binding force, and continue to
express the form in which, if we could give it to the whole
material of thought, our mind would find all its demands
satisfied. This form also, therefore, has a right to its place
in the systematic series of forms of thought : that it is the
last in the series is clear without proof, for it leaves no
elements remaining in mere unconnected juxtaposition, but
exhibits everything in that coherence which had been all
along the aim of thought. At the same time it points
beyond the province of logic. From the point of view
of the explanatory theory it might still seem as though the
universal laws, which thought produces from itself alone,

i$S THE THEORY OF INFERENCE.

gave a right to decide a priori what reality will be like ;
speculation does not deny this right, but by making the
content of a supreme principle the one and only ultimate
ground of everything, both of the power of these universal
laws themselves, of the direction in which the world as a
whole develops, and of the individual forms which in con-
sequence reality assumes at each moment, it indicates that
the final fulfilment of all logical aspiration could not be
attained by new logical forms, but only by material knowledge
of that supreme self-developing principle which speculation
presupposes.

In concluding this account I am conscious how much its
method deviates from those which are in vogue at the
present day. We are so accustomed to being told the
history of things, and to feel our curiosity satisfied when we
have discovered or invented an origin for them, that even
logic is flooded with psychological explanations and deriva-
tions of its doctrines : on the other hand it strikes us as
antiquated, odd, and unmeaning if anyone attempts to
arrange the forms of thought in a progressive series
according to the nature of its problems, instead of following
the order in which the mental activities necessary to their
solution develop in the individual soul. I am content that
this should be so, and hope that in the form of my ex-
position my readers will recognise the premonitory influence
of the idealistic philosophy to which it is intended to lead :
I have no fear that by choosing this form I have distorted
the substance of truths which, on any view of logic, must be
equally regarded as established.

BOOK II.

APPLIED LOGIC.

PREFATORY REMARKS.

152. WE are so much accustomed to oppose the world
of our thoughts to an external reality, that as soon as we
speak of an object to which the forms of our thinking are
to be applied, it seems as if we can mean thereby nothing
but this external reality. When we call to mind the natural
sciences, which occupy so large a portion of the field of
science at the present day, we are confirmed in this opinion ;
on the other hand, when we think of mathematics and
jurisprudence we are likely to be shaken. The external
reality supplies neither the objects with which the mathe-
matician^ deals nor the methods by which he deals with
them. That which it yields does but give him an occasion
to turn his investigations in this or that direction. The
true objects of his enquiry are always nothing but the forms
which our intuition or our thinking finds in itself or creates,
and of which the appearances of the outer world remind us,
without ever perfectly corresponding to them. And his
business is, in accordance with laws of reasoning, which
at any rate are not derived from any external experience,
to develop the countless necessary conclusions which follow
from the various possible combinations of these forms.
Nor is this development speedily achieved : these con-

200 PREFA TOR Y REMARKS. [Book II.

sequences do not unfold themselves in such a way that
we need but to look on and watch : on the contrary logic
has at all times turned to mathematics (for the two are
coeval) for examples of delicate profound and fruitful
methods of enquiry.

Jurisprudence certainly owes the occasion of its origin
to the circumstances of the actual world in which man with
his needs and claims is placed; but it tries to shape this
world and our relations to it by ordinances, which, though
as against nature they are products of our free choice,
are yet the necessary consequences of ideas of right and
justice, consequences of a truth that ought to be, which
has its home nowhere but in our own minds. And so
logical acumen is just as constantly employed here also
in setting forth ever more precisely and irrefragably the
connexion of the several conclusions already drawn both
with one another and with the highest principles from which
they flow.

Thus both these branches of science show that logic
need not go to the external reality to find objects for its
application, that it finds fully work enough in investigating
the connexion of that which is possible in thought and
necessary in thought, that finally the inner world of our
conceptions is wide enough to contain unknown regions,
still to be discovered by means of systematic enquiry.

153. Keeping to this line of thought we may now turn to
the natural sciences. Even the external world which we
assume is after all an object of our enquiry only so far
as (in some way or other which does not here concern
us) it has become a world of conceptions in us ; we survey,
dissect, and investigate not that invisible something which
we suppose to lie outside us, but the visible picture of
it that is formed in our consciousness. We may believe
that we are compelled, as the result of prolonged labour,
to accept certain connexions according to law between the
unknown parts of this unknown external something ; but

Book II.] PREFA TOR Y REMARKS. 201

all these assertions (whatever they may be) are after all
grounded solely upon the relations which prevail either
persistently or in succession between the contents of our
thoughts. Whatever may be the causes which produce
this succession, the laws by which it is regulated can
only be known by itself, i. e. by the order in which certain
thoughts follow certain others in our minds, by the constant
union of some thoughts, and the impossibility of uniting
others. It is enough then even for the treatment of the
external world to regard it in the first instance as a world of
thought set up somehow or other in us ; whether the ap-
pearances which surround us correspond to a real world
of external things, or whether they be products of a creative
faculty of imagination in us, guided by unknown impulses,
the discovery of the connexion between them will always
necessitate the same methods of enquiry.

I wish the reader to bear in mind what I have said
as we pass to applied logic. My purpose in saying it here
is only to indicate the position taken up in the following
enquiries : in the course of these enquiries we do no
violence to the ordinary way of thinking; let the reader
while he reads these chapters conceive of the efforts of
thought as directed to a real external world ; only when
he finds no notice yet taken of the relation of this world
to our thought, I hope he will find a justification of this
course in these few prefatory remarks, and be content
to wait till the third part of my treatise for an enquiry
into the significance of the issue which is here put aside.

CHAPTER I.

The forms of Definition.

154. INNER states, sensations and ideas, feelings and
impulses, cannot be conveyed like material things, which
may be separated from their original possessor and passed
on as they are from hand to hand. We can communicate
them only by subjecting our neighbour to conditions under
which he will be compelled to experience them or to beget
them anew in himself.

If we had to communicate for the first time something
yet unknown, which was too simple to be created by
thinking, or too complex to be exhausted by it, our only
resource would be to produce the external conditions of
perception. If our neighbour had never seen light, or
heard sounds, or felt bodily pain, our only course would
be to put his eye within reach of a source of light, to bring
waves of sound to act upon his ear, and by the application
of a stimulus to his body to let him experience that feeling
of pain with which we ourselves had made acquaintance
in precisely the same way. If we wish to enable him to
recognise a person whom he as yet does not know, the
description of the countless little marks which distinguish
that person from others will never make sure, but by
pointing with the finger we can show him precisely whom
we mean. We need do no more than thus barely mention
the fact that wherever it is applicable this direct reference

CONDITIONS OF COMMUNICATION. 203

to the object itself or to some likeness of it is always
useful. But in view of the questions which here concern
us we further presuppose two things, first a large stock
of past experiences common to the persons who are to
communicate with each other, and secondly a language
intelligible to both parties, to the several words of which
each attaches (to a large extent at least) the same ideas.
Then by a series of spoken words we call to our neighbour's
recollection the ideas conjoined with them in that order
which is for him the internal condition of his creating or
experiencing in his own consciousness that which we wish
to communicate.

155. This form of communication also includes much
else that our logical enquiry can only take note of by the
way. Both poetry and eloquence aim by this method at
something more than imparting ideas : they count upon the
attachment to the images thus called up of feelings of
pleasure and pain, of approval and disapproval, of exaltation
and aversion. The effects which they thus produce are
powerful but uncertain. Different minds are indeed pretty
uniformly organised for the mere apprehension of matters
of fact, and their general habits of perception do not
change; but in estimating the degrees of emotion which
we annex to what we perceive we must allow not only
for original differences of temperament, but also for
the changefulness of the mood of the moment, which
depends upon what we have just gone through. Thus
different persons are very differently receptive even of actual
facts ; still less can we hope by the imperfect recollection
of such facts, which is all that speech can rouse, to create
in others precisely the same emotion which they produced
in ourselves. How much may be done by skilful guidance
of the train of ideas and by well-measured expressions to
lessen the uncertainty of the result is a question for the art
of poetry and rhetoric. Our own problem is narrower and
is limited to the communication of that which has been

204 THE FORMS OF DEFINITION*. [Book II.

already refined from a state in which we are acted upon
into an idea which we apprehend, i.e. of thoughts, not of
feelings and moods.

156. The certainty even of this kind of communication
seems to be imperilled by the fact that after all the same
words do not always have the same meaning for the speaker
and the hearer. It must be allowed that, apart from
subsequent confusion of originally different roots, there are
in every language many words which denote several very
different things, in consequence no doubt of a resemblance
which these things bear to one another, but still of a
resemblance which is not always so obvious now to him
who uses the traditional words as it was to the first inventor
of these metaphorical expressions. And even when a word
denotes the same thing for all, that does not ensure that all
have the same conception of the thing denoted. The
special circumstances under which each individual became
acquainted with the thing, the peculiar point of view from
which he first regarded it, the connexion in which he found
it and from which he had to detach it, give a peculiar
colouring to his picture of it, and dispose him to other
conclusions than those anticipated by the speaker when he
named the common word, hoping thereby to give some
particular turn to the course of his hearer's thoughts. It is
impossible to deny these facts, dangerous to disregard them
altogether, yet foolish to press them too far : the intercourse
of daily life sufficiently proves to how large an extent speech
enables us in spite of them perfectly to understand each
other's thoughts about the most various matters. There
will certainly remain ideas which it is hard to communicate
with precision ; but were there no such difficulties there
would be no good in seeking rules for helping us by the
appropriate use of unequivocal words to remove the ambi-
guity of others and to fix their meaning so that all who
wish to converse may use them in the same sense. It
must be left to the unfettered acumen of the speaker to

Chap. I.] DEFINITION BY ABSTRACTION. 205

determine what words may be accepted as precise enough
to explain other words ; but however far we may feel con-
strained to go back along this line and to remove all
ambiguity from the instruments of communication which
we wish to use before we use them, there will still be only
two possible ways for us, abstraction and construction.

157. We explain a conception, which we will call M 9 by
abstraction, when we first refer to a number of known
instances, in each of which M forms a part of the notion,
and then bid the hearer separate from these instances that
which does not belong to the conception M which we wish
to communicate. This is the way in which all our general
conceptions 1 and general ideas 2 were originally formed; in
the case of a general idea that which was common to a
number of impressions comes of itself t stand out as the
object of a new separate idea ; in the case of a general
conception this process is consciously directed by attention
and reflexion. And when we are at a loss we all come
back to this same way. The man of no logical training
does so when to the question what he understands by M he
replies, in the fashion which the Platonic Socrates so often
complains of, only by giving examples which contain M y
leaving to his questioner the trouble of separating the
common element which he wants to get at from that which
is foreign to it. But the logically trained thinker also
proceeds really in the same way: however carefully he may
choose his terms so as to express the universal itself with-
out any reference to particular instances, yet this expression
is only obtained by a tacit comparison of a number of cases.
It is only by such a comparison that we learn what marks
of M must be precisely fixed in order that the expression
may exclude all that is foreign to M, what other marks
must be left undetermined in order to include in M every-
thing that is properly an instance of it. And lastly, only
by the fact that instances are to be found are we convinced
i [' Begriffe.'] a [' Vorstellungen.']

206 THE FORMS OF DEFINITION. [Book II.

c

that this My which we are taking the trouble to determine,
is capable of determination, that it represents a problem
which has an intelligible solution, not a mere tissue of in-
compatible elements whose union may be demanded in
words but cannot be really carried out.

158. It is thus useful to follow this method of abstraction
in every case, and even when we may have arrived at a
determinate conception in some - other way, at any rate to
confirm it by a supplementary reference to instances.
Wherever our aim is to fix some very simple conception
which underlies a whole group of kindred ideas, it is the
only method possible. Such a conception can only be
pointed out by taking away from known instances of it all
that does not belong to it ; we can never put it together
out of its component parts, for it has none. The labour
expended upon this impossible aim always ends in a vicious
circle, since among the materials that are to be used in the
construction the very thing that was to be constructed is
taken for granted, whole and entire, however much it may
be concealed under strange expressions. Thus, for example,
in our idea of becoming the two ideas of being and not-being
are no doubt united as two connected points of relation ;
but if we should try to characterise becoming as the unity
of the two we should not attain our object. In the first
place we should be bound to fix the precise sense to be
here assigned to the expression ' unity ' which in itself is
very ambiguous. It cannot mean the mere co-existence in
the same consciousness of the two ideas of being and not-
being, for obviously becoming is the content of a relation
that exists between the contents of these two ideas. But if
we try to unite being and not-being as predicates applicable
at the same time and in the same manner to one and the
same thing, we do not arrive at becoming, but simply find
ourselves confronted by the impossibility of actually exe-
cuting in thought a task which involves such a contradiction.
Suppose then that we separate again the being and the not-

Chap. I.] DEFINITION BY CONSTRUCTION. 207

being of this thing and say that the one predicate is applic-
able to it when the other is not : even by this change we
do not get hold of becoming; it falls between the two
moments of time and is to be found in neither. We shall
have therefore to bring them together once more : but as
long as they are separate from one another becoming will
lie outside of them, we can only get hold of it when we
look for it neither in being nor in not-being, nor in a passive
unity of the two, but in the transition from one to the other.
But in this idea of transition, or in any idea however it be
expressed that we like to substitute for it, we shall recognise
(only under another title) what is essentially our idea of
becoming. This relation therefore between being and not-
being, as it is altogether sui generis, cannot be conceived by
means of anything but itself, is only to-be got by abstrac-
tion from the instances in which it forms a part of the
thought, not to be created by the putting together of ideas
which as yet do not contain it. Precisely the same con-
siderations hold with respect to the equally simple concep-
tions of being, acting, thinking, affirming, denying ; and the
geometry of Euclid follows precisely the same method in
determining the surface as the limit of the space occupied
by a body, the line as the limit of the surface, the point as
the limit of the line, in each case teaching the learner to
get the simpler conception, which is harder to grasp, by
abstracting what does not belong to it from the more com-
plex conception which lies nearer to sense or which has just
been determined.

159. The opposite method would fully deserve the name
of construction only if it enabled us completely to put to-
gether the idea to be conveyed out of a definite number of
unequivocal parts by a series of acts of thought which we
were required in unambiguous language to execute upon
those parts. Almost the only conceptions that really admit
of this treatment are the mathematical conceptions and
some others that arise out of the applications of mathe-

208 THE FORMS OF DEFINITION. [Book II.

matics, conceptions which as creations of our thought
contain only what our thought has combined in them.
They admit of it because the several ideas which make up
the whole conception can be completely enumerated, and
because not only each of these ideas but each of the ways
in which they are to be joined together is such that we can
state the characteristic quantity by which it is distinguish-
able from others of its kind, as well as the special quality
which distinguishes it from those of another kind. Here
then nothing remains indeterminate that should be deter-
mined ; he who follows the directions given must see the
picture he is desired to form rise before his mind's eye with
just that degree of individuality or generality which the
speaker wished to give it.

If on the other hand we wish to convey a notion of some
really existing thing we are met by well-known difficulties.
Our mental picture of a real thing is not made up of a
limited number of points of relation which are to be brought
into combinations also limited in number, but is com-
pounded of a countless number of ideas. And of these
component ideas those that belong to different senses can-
not be compared with one another, while even those of the
same sense can only be designated by general names, and
scarcely admit of precise measurement. And lastly it is
beyond our power to make a complete survey of ..the com-
binations of all these elements, nay we cannot perceive
them at all except so far as they consist of an external
arrangement in Space and Time, and even then we cannot
find any comprehensive expression for them in our ignor-
ance of any pervading law of their formation.

In the presence of this fulness of detail construction
shrinks into description. In describing we try, if we under-
stand our business, first to fix the main outlines of the
whole idea, whether this be done by a simple construction,
or by taking as illustrations similar things already known
and proceeding by alteration and transposition, by the

Chap. I.] GEJ\'US AND DIFFERENCE. 209

removal of some features and the addition of others, to
elicit from them the leading lines of the picture we wish to
convey. Then we fill in the mass of details, never com-
pletely, for they arc usually inexhaustible, but skilfully
selecting those by the mention of which we may hope that
the hearer's attention will be at once stimulated to supply
from his own memory those that are not mentioned. We
need but remind the reader of the wonderful effects which
the poet produces in this manner, bringing a whole picture
before us with a touch ; though the uncertainty of the result
is equally manifest. The way in which each man supplies
what is not mentioned vanes according to his nature : were
it possible to bring to view in detail the different pictures
which the same description calls up in different hearers,
their variations would show what an inadequate basis a
description must be for the support of definite conclusions.
For scientific purposes therefore description needs a regula-
tion of its method, and this it finds in the rules of defini-
tion.

16O. For the definition of a conception M it is usual to
require a statement of the next higher generic conception
G (the genus proximum\ and of the characteristic mark d
(the differentia specified) by which M is distinguished from
other kinds of G. By requiring the generic conception G
we set bounds to the arbitrary and capricious course of
description. In describing you were free to begin at any
point whatever, and then gradually to add the remaining
points in any line that you pleased, so long as you could be
sure of producing in the end a clear picture of what you
meant But even in a description you would not attain
your end without the employment of many general con-
ceptions. Now instead of an arbitrary choice of these, the
rules of definition require you to start from that universal
conception in which the largest part of the constructive
work before you lies completed and ready to hand, and
which, being denoted in speech by an unequivocal name,

LOGIC VOL. I. P

210 THE FORMS OF DEFINITION. [Book II.

may be assumed to be familiar to every mind, fitted to serve
as the outline for the filling in of the details by which the
intended picture is completed.

If we are told that a creature we have never yet seen is a
bird, this general conception gives us at once a clear picture
of a number of members united in a characteristic manner,
and at the same time of the peculiar kind of locomotion
and vital action to which they are instrumental. The
further special characteristics are easily added to this out-
line, for it indicates of itself the places to which they
severally belong. We should never get such a clear idea of
the unknown creature if we had to put it together out of its
primary components. It would be an endless task to
enumerate all the variously-coloured spots on its body with
their position and the extent to which they may be dis-
placed, so as to give a notion even of what it looks like.
Still more endless would it be to add to this the peculiarities
of life and habit, which all belong at any rate to our idea of
the animal in question if not strictly to our mental picture
of it.

We see then the value of the abbreviation effected by
starting from a general conception that can be assumed as
known : we understand also that we must choose for start-
ing-point not merely any higher universal, but expressly the
genus proximum, which in its characteristics and in the
mode of their combination comes closest to the conception
to be defined, and so clearly describes the point at which
and the manner in which we are to add each of the last
characteristics by which the conception is finally determined.
By starting from a higher universal than this we should not
only lengthen again the rest of our task, which definition
was intended to shorten, but we should run a risk of failure.
For we should then have to add a whole series of further
characteristics in order to exclude everything foreign in the
long descent from that less determinate universal to the
particular species in question : and each new characteristic

Chap. I.] THE SPECIFIC DIFFERENCE. 211

would open a new source of error ; for it is hardly possible
to determine quite precisely the mode and manner in which
each is to be added to those that have preceded it without
appealing to a picture which it may be assumed that each
man already has in his mind. The notion of that genus
proximum therefore would not by this method be produced
afresh with that definiteness and certainty with which it
could be recalled to the memory at once by the mention of
its name, and which it must have if it is to serve as an out-
line for the filling in of the final characteristics of the con-
ception which we desire to convey. All that we could get
by this method would be more or less of a riddle. For
when we propound a riddle what we do is this, we tell our
hearers without more ado to attach to a very indefinite
universal (a mere something that may be anything) predi-
cates that can be united only in one very definite subject,
leaving it to his ingenuity to find this subject or in the first
instance the genus proximum which admits of their union.

161. As yet we have spoken of the definition as a me-
thodical description. If it is to retain this character it would
have with regard to M to state completely the modified
forms p 1 <? 1 r l assumed in the case of M by P Q J? the
general predicates of the genus G. Instead of all these
characteristics the usual rule for definition requires us to set
down on!y one characteristic d, the specific difference, by
which M is distinguished from all other species of the
genus G, Definition thus has a more limited and therefore
a more practicable aim than description : instead of setting
forth positively the whole content of M it has only to state
the mark by which M may be separated from all that is not
M. This is the origin of the terms dcfinitio and opines,
both of which imply only the marking off of one thing from
another. And in fact the general aim of definition must be
thus limited. As thought advances we feel no doubt the
need not only to distinguish, but to know completely what
we have distinguished.; then we make further demands

p 2

2\2 THE FORMS OF DEFINITION. [Book II.

upon definition ; then we refuse to admit as a specific
difference anything but one of those characteristics that
really make a species, i. c. one whose occurrence decisively
modifies the forms assumed in M (the thing to be defined)
by all the other characteristics of the genus G which are not
mentioned in the definition. These heavy demands how-
ever can be completely satisfied only at the conclusion of an
enquiry which has made us perfectly acquainted with the
nature of Jlf, and which thus enables us to solve the problem
which remains, of fixing a final and classical expression for
that nature.

But besides this there are other no less pressing problems.
We may have to begin a speculative enquiry, which has to
find a number of yet unknown propositions that are true
of M } or in a practical matter we may have to determine
what is the proper consequence of a given situation M: in
either case it is of the utmost importance that this M, to
which the propositions we are going to assert or the decision
we are going to arrive at must apply, should be marked off
by precise and easily traceable boundaries, nay at first this
is the only thing that is of importance. For this purpose
any characteristic d will suffice, even the most insignificant,
provided only that it be really an exclusive mark of M. In
the first case, that of a speculative enquiry, the further
course of the enquiry itself will either reveal the reason
which connects the validity of a series of propositions with
the presence of this obscure characteristic d, or will show
that they are valid over a wider or narrower field than this,
so that d is not the proper characteristic of their subject. In
the other case, that of a practical matter, the exact meaning
of a legal situation to which a law is to apply must be
completely considered beforehand while the question is still
de legj ferenda ; but he who has to carry out the lex lata
rightly demands that this previous consideration shall have
given the law the form of a definition which distinguishes,
not by the most profound but by the most obvious mark,

Chap. I.] NOMINAL AND REAL DEFINITION. 213

the cases to which a decision shall apply from those to
which it shall not. These are problems which applied logic
cannot decline, and we overlook them when we think too
disparagingly of this traditional form of definition. We
misunderstand the sound sense of many such definitions in,
practical philosophy and jurisprudence when instead of the
marks of M, which they intend to give and do give com-
pletely, we see in them nothing but an inadequate statement
of the whole nature of -#/, which it is not their purpose to
give at all.

162. It will be convenient to notice in this context the
distinction which is commonly drawn, but not always in the
same sense, between nominal and real definitions. We may
utter a name or replace it by another; but we can never
define anything but its meaning, i.e. our idea of that which
it is intended to signify: the thing itself again is not in our
mind, but only the picture we have formed of it. These
two kinds of definition therefore seem to be identical ; and
they are in fact identical for everything that exists only in
our minds, and whose whole nature therefore is exhausted
by our idea of it. There is no real definition of a geometrical
figure that can be distinguished from its nominal definition ;
any correct definition that we give of it expresses at once
the whole nature of the thing in question, and the whole
meaning of the name.

In other cases however the distinction between these two
modes of definition is one that it is worth while to make.
If we call the soul the subject of consciousness, of thinking,
feeling, and willing, this may be appropriately called a
nominal definition ; it specifies a condition which a real
thing must satisfy if it is to be entitled to the name of a
soul. But who or what this thing is whose peculiar nature
enables it to satisfy this condition, is still quite an open
question ; we have not fixed the real definition of the soul
till we have got a theory which proves either that only a
supersensuous and indivisible being, or that only a con-

214 THE FORMS OF DEFINITION. [Book II.

t

nected system of material elements can be the vehicle of
consciousness and its various manifestations. It was a
nominal definition of beauty that Kant gave when he said
that it is to be found not in the conformity of the beautiful
object with some conception, not in its capacity to satisfy a
desire in us, but in the fact that it pleases directly and
without reference to any interest. The real definition of
beauty would have to point out the precise relations between
various things or components which enable every object in
which they occur to produce this pleasing effect. And so
we may say in general terms, when experience shows us a
group of characteristics p qr often occurring and continuing
together, or when in the course of our investigations we
light upon a coincidence which induces us to put them
together and to regard the group as a subject for further
enquiry, we proceed in the first instance to form for the
group a conception M, of which a nominal definition can
always be given, because it has only to set forth the predicates
which led us to invent the name, or the effects which we
expect from the thing to which the name is applied. But a
real definition cannot always be given : for there is no
assurance that we have not combined in M characteristics
whose union we thought ourselves justified for some reason
or other in assuming or desiring, when there is in fact
nothing to be found in which they really are or can be
united. It is a common error to mistake this mere indica-
tion of a problem we should like to solve for the solution
itself; and on this account the distinction between these
two kinds of definition is useful as a warning.

163. We have to beware of three faults which vitiate
a definition.

In the first place its assertion M=Zmust be no tautology ;
but it becomes a tautology whenever M itself is explicitly
or implicitly assumed among the ideas combined in Z by
which M is to be explained. This fault (called drculus in
definiendo) is often committed through carelessness which

Chap. I.] FA UL TS OF DEflNl TION. 2 1 5

no rules can prevent ; but we are almost of necessity driven
to it whenever we try to give a formal definition of
some simple thing which does not fall under any more
general conception.

In the second place a definition, since it has to fix a
conception, must be a universal proposition, true of every-
thing which falls under the conception. Now if every
J/=Z, it follows by contraposition, that no M is not-Z:
if then further reflexion or fresh experience teaches us
that after all there are some M which are not-Z, we know
that the definition M~Z was too narrow (definiendo
angustior) and was not, as it ought to have been, true of
every M.

Lastly a definition must be convertible : if every MZ,
it must also be true that every Z J/: whenever therefore
further reflexion or fresh experience shows that some Z
are not M, we know that the definition MZ was too
wide (definiendo latior)^ and included some non-J/ which
it ought to have excluded.

To point out how to avoid these faults would be more
useful than thus merely to name them ; all we can do
in that way however is to indicate their usual source,
viz. the limited range of our observation, which as a rule
opens to each individual only one and the same fragment
of the entire field covered by a conception, and further the
one-sidedness into which our thinking is apt to lapse if
it does not constantly receive fresh stimulus from without.
In the temperate zone the way in which plants awake in
summer and sleep in winter makes a strong impression
upon our feelings ; animal life, with its continuous activity,
seems to offer a complete contrast. Now we certainly
should not base upon this a scientific distinction between
animal and plant ; yet countless comparisons, employed
by poet and orator, show that we are accustomed to con-
sider this yearly alternation as the essential characteristic
of the plant. But a definition which expressed this would

216 THE FORMS OF DEFINITION. I Book II.

be at once too narrow and too wide : it would exclude
tropical plants whose life is an uninterrupted growth, and
would include hibernating animals, which in this climate
easily escape our attention, directed as that is mainly to
the domestic animals. It may easily happen that one who
wishes to establish on a new basis the rights and duties,
both political and social, of all the members of the state,
thinks only of the male world to which the conduct of
these transactions is usually confined, and then his pro-
posals will be too wide, in as much as he demands for
all what he intends for men only, or too narrow, in as
much as he expressly enacts for men only what must
obviously apply to all. From this we may draw a lesson
of universal application : we should never attempt to treat
a problem off-hand, when it is possible to extend the
limits of our own experience by converse with others or
by taking count of views which are already recorded in
the literature of the subject. Learning is not in itself
inventive, but like any other training and discipline, it
makes us more secure against extreme errors than if we
proceed by the mere light of nature.

164. We further require in a definition elegance and
brevity, which I will illustrate by a simple instance. If
we define a circle as a curved line all the points of which
are equidistant from its centre, we first of all make an
actual mistake in giving too wide a definition. For if on
the surface of a sphere we draw a serpentine line which
crosses and recrosses a great circle of the sphere making
equal curves on either side, all the points of this line
are equidistant from the centre of the sphere.

If further the line, in returning to its origin in the great
circle, describes an uneven number of these double curves,
it will consist of an infinite number of pairs of points, form-
ing the opposite, extremities of so many diameters of the
sphere. The centre of the sphere therefore bisects the rec-
tilinear distance between the two points of each pair ; and

Chap. I.] ELEGANCE AND BREVITY. 217

i

so, in every sense which can here be given to the word, it
would also be the centre of the sum of all these pairs, i.e.
of this line, which nevertheless would not be a circle. We
ought therefore to have said that a circle is a curved line in
one plane which fulfils the above condition.

But elegance further demands that a definition shall not
contain more ideas than are indispensable for the complete
determination of the given conception. So we may be
called upon to speak not of a curved line but of a line
simply: if a line fulfils the annexed condition it follows
without more ado that it cannot be straight. The condition
itself however is not correctly expressed. A definition
should not employ among its instruments of explanation
ideas which are themselves unintelligible without the con-
ception to be defined. In this case the idea of the centre
is certainly such an idea. If we had not yet got the idea of
a circle (and in fact there is nothing in this case at least to
suggest this idea to us, after we have omitted the character-
istic of curvature from our definition) we could at first think
of the centre of a line only as the point of bisection, and
we should not discover our error till w r e attempted to con-
struct a circle on that understanding. Instead therefore of
this sense of the term centre which common usage suggests,
and which compelled us to be so painfully discursive just
now in speaking of our serpentine line, the definition re-
quires the precise statement in general terms of the meaning
which the word is to bear for all figures whatsoever. This
statement can easily be given, but I may omit it, as it
follows therefrom that //there be a point in a plane which
is equidistant from all the points of a line in that plane, that
point is the centre of the line. But if we now introduce
this definition of centre into our definition of a circle, the
statement of the further condition under which the line in
one plane becomes a circle comes to be a mere tautology,
and the meaning of the whole definition is evidently nothing
more than that a circle is a line in one plane for which

2l8 THE FORMS OF DEFINITION. [Book II.

there is a point in the same plane from which all its points
are equidistant. The definition is substantially correct ;
yet fault may be found with its form. For now after
omitting the term centre we remember that it was only the
presence of that term that forced us to look for the equi-
distant point, in the same plane. Not this actual centre
only, but any point in an axis drawn through it at right
angles to the plane of the line fulfils the condition of being
equidistant from all points of the line. It is enough there-
fore to say that a circle is a line in one plane such that a
point may be found from which all its points are equidistant.
It is needless to mention that there are several such points
and to say where they lie : the attempt to construct the line
according to this direction will at once teach us both. But
once more even in this form the definition is not quite all
that can be desired. It does indeed say that all the points
of a circle are equidistant from one and the same point, but
it does not formally state whether or no all points that are
equidistant from this point are points in the circle. They
are so in fact provided they he in the same plane, and thus
in order to express this along with the rest we may finally
say that a circle is a line which contains all the points in one
plane which are equidistant from any point.

165. Different opinions may be entertained as to the re-
quirements of definition which I have just illustrated by the
example of the circle. Every one will allow that it is a
serious fault to employ ideas which (like centre in this case)
though a meaning may be given them apart from the con-
ception to be defined, yet are not fully intelligible without
it, except perhaps in the context of a scientific treatise. But
it may be thought that the addition of superfluous cha-
racteristics is unobjectionable, since it makes the definition
easier to understand without impairing its correctness.
Nevertheless it should be avoided. For the addition of
some characteristic z that might be dispensed with, is apt,
as we are not told that we might dispense with it, to make

Chap. 1.] DEFINITION OF ADJECTIVES $ VERBS. 219

us think that it is inserted in order to distinguish the M we
are defining from a non--#/to which everything in the de-
finition is applicable excepting only z. If we say a circle is
a curved line in one plane such that there is a point from
which all its points are equidistant, the form of the state-
ment suggests that there are also straight lines which satisfy
that condition. It matters little in so simple a case as this ;
but in more complex cases serious disadvantage may be the
result of this apparently harmless addition of superfluous
matter. At the least it hampers us in the drawing of con-
clusions, which after all was our sole purpose in laying down
the definition. It may happen, for instance, that it has
been quite clearly established, perhaps in some indirect
way, that Q has the whole sum of predicates that are suffi-
cient according to the correct definition for the subsumption
of Q under M, but that it is difficult or impossible to prove
directly that Q also has the predicate z which is superfluously
added in the definition actually given : there will then be a
quite useless hesitation about bringing Q under M and
actually drawing the conclusion which that would justify.
And so we may say generally that it is right to demand
that a definition shall contain only those terms that are
indispensable for the specification of the object, but shall
exclude all merely descriptive elements : if it does not
enable us very readily to form a picture of the thing, this
will be atoned for by the certainty of the conclusions we
can draw from it.

166. Hitherto we have been considering the usual form
of definition by the proximate genus and the specific
difference as the only valid form. But the untrained in-
tellect is wont, to the annoyance of the logicians, to use
another mode of definition, and to say, for instance, in its
familiar uncouth way, sickness is when something pains me.
Such a phrase certainly needs to be amended, yet not
exactly in the way which logicians rather intolerantly
require, but rather in the way in which physical science

220 THE FORMS OF DEFINITION. [Book II.

actually defines many of its conceptions. The ordinary
form is properly adapted only for defining the meaning of
a substantive : when we have to do with adjectives and
verbs it is not only shorter but more correct to give them
their proper place in the grammatical structure of the
definition, and to let them bear plain reference to their
subject, seeing that it is only as expressing states or pro-
perties of a subject that they have any meaning. It is quite
right therefore to define adjectives like sick or elastic by
such propositions as 'a living organism is sick when its
functions depart from a certain course;' 'a body which on
the cessation of external constraint resumes its original
shape is elastic. 1 And in defining the meanings of the
verbs to live and to sin it would be quite proper to name
first the subjects to which they can be applied, an organic
body and a spirit that is conscious and wills, and then the
conditions under which they are to be predicated of these
subjects. It is absolutely useless to begin by throwing all
these ideas into the substantive form and ranking them
under the head of states or properties or modes of action :
that they are to be so ranked is at once apparent if we
leave them their adjectival or verbal form and give them
their proper place in the sentence. The usual mode of
definition on the contrary has the disadvantage of making
us far too apt to separate from its subject and treat as
independent what is nothing but a state or property of
something else. When we have once framed the substan-
tives sickness, sin, freedom, it is hard to keep quite clear
of the strange mythology which speaks as if these terms
stood for things with a being of their own, and traces their
development, without ever seriously coming back in the
course of its enquiry to their real subjects, though it is only
as properties, states, or activities of these that they exist,
and though their apparent development is every moment
bound up with the real development of these subjects.
167. Under the head of conceptions to be defined we

Chap. I.] GENETIC DEFINITION. 221

have hitherto considered only comparatively simple ones,
conceptions of figures, things, properties, and easily in-
telligible relations : but among the words used in speech,
every one of which may under certain circumstances call
for a definition, we often find very complex relations
between a great variety of points of attachment com-
prehended in one simple expression. No one who was
not hide-bound by prejudice would require that the ex-
planation of such conceptions should take the regular form
of a simple definition ; and to find special names for all
the other very various methods which may be employed
would be nothing but useless pedantry. The universal
principle of applied logic is simply that all ways are
allowable which lead to the goal ; it hopes for no more
than to remove our doubts as to which way is passable
right up to the end, and which not, by pointing out that
w.hich has long ago been tested : it never forbids our
seeking new ways to satisfy new needs. It is always
allowable therefore to begin with a preliminary description,
with comparisons and analogies, with discussions of any
kind, in order to familiarise the hearers with the meaning
of the subsidiary ideas we wish to employ and the peculiar
combinations we wish to establish among them, and having
thus prepared the way to proceed to set forth what we
wished tp explain in a formula which is brief and intelligible,
though it presupposes what has gone before and cannot be
separated from it.

This reminds us however of another twofold division of
all definitions. We may characterise M by the aggregate
of marks displayed by the conception when it is present to
our minds in its completeness : this kind of definition,
which we illustrated just now in the case of the circle, may
be called descriptive definition : we have recourse to it
mainly in the case of actual things which we only know
from the outside and whose definition therefore is in fact
nothing but a methodical description. But we can also fix

222 THE FORMS OF DEFINITION. [Book II.

c

M by pointing out a way in which, riot by the mere addition
of other ideas, but by freely using and manipulating them
at will, this idea can be produced with certainty. This I
would call genetic definition, understanding thereby (and
this I wish particularly to emphasise) not a statement of
the process by which the content of the conception M is
actually found, but only an indication of the way in which
the mental picture of this content M may or must be formed.
' Let a straight line revolve in one plane about one of its
extremities, and combine the successive positions of the
other extremity:' that is a genetic definition of a circle.
The circle as such is not made at all : but supposing a
particular circle such as we draw to have been already
made in some way or other, we may certainly form a
mental picture of it in the way indicated by this definition.
But we may form that mental picture equally well by sup-
posing the length of the two axes of an ellipse to alter till
both are equal to r ; or by supposing a cone to be inter-
sected by a plane at right angles to its axis. And thus an
idea, whose content has in itself no genesis, may admit not
only of one, but of so many genetic definitions as there are
ways of forming the idea of this content by the manipula-
tion of other ideas. Among these genetic definitions then,
using the term in a somewhat extended sense, we may
include the above-mentioned miscellaneous methods : they
try by indirect means to make us form a mental picture of
M) when it is impossible or inconvenient to say directly
what M is.

168. Strictly speaking, whenever we undertake to define
a conception J/, our aim is to give it a higher degree of
definitcncss than it has yet. But in fact the problem
usually narrows itself to the transformation of a clear idea
(clara perceptio) which we already have of J/, into a distinct
one (distincta\ or of a mere mental picture, which does but
comprehend M in a loose general way as a connected whole
made up of parts which are familiar, into a real conception of

Chap. I.] CLEAR AND DISTINCT IDEAS. 223

M. These two expressions may be regarded as equivalent.
For according to old established usage we are justified
in saying we have a clear idea of anything when we think of
it as one, and as a connected whole, and lastly as dis-
tinguished from others with precision enough to avoid
confusion; but it does not become distinct till to this
is added the general law which regulates the connexion
of the parts, and further the characteristics which it has
in common with other species of a certain genus, and lastly
those particular characteristics which distinguish it from all
the other species of its own genus. In treating of Pure
Logic we identified this increase in definiteness with the
transition (in technical language) from an idea or mental
picture to the conception or actual comprehension of a
thing.

But now there are cases in which our idea of an M which
is to be defined is far from possessing the clearness here
supposed : names are handed down to us which have
become part of our language though their meaning has
never been precisely fixed. Thus we speak of virtue and
sin, of good and the highest good, of appearance and
reality, with a full conviction that we mean something
very definite by these names, and ready to draw important
inferences from them in reference to that to which we
apply them. But at last the difficulties in which we en-
tangle ourselves convince us that strictly speaking we did
not know precisely what we meant, that we had not com-
pletely fixed the conditions which must be satisfied in order
to justify the application of these names, that we had in
short trusted to hazy ideas, the clearing up of which
is of the very first importance. This we try to effect
in a very simple way. If we were entirely ignorant of the
meaning which M was intended to* bear, we should have
no means of finding it out ; but also it would never have
occurred to us to apply this name had not some part of
its meaning (say a) been fixed beyond a doubt that very

224 THE FORMS OF DEFINITION.

t

part namely which now impels us to use the term the rest of
whose meaning is still hazy. This a we first take tentatively
as a complete definition of J/", and consider whether a cor-
responds to what we mean by M. It is a matter of common
experience that in cases where we are not in a position
to express the meaning of M in positive terms we may yet
see whether an idea a that is offered as a definition of
it is adequate or not. Thus when we are trying in vain
to recollect a name we can yet pronounce with perfect
certainty that a suggested name is not the right one ; and
further any resemblance it may have to the right one
makes an impression on us, and sometimes reminds us
at once of what we want, at any rate it helps to make
plain the other points in which the right name differs from
the suggested one. We are in the same case here : a is not
utterly wrong and incapable of comparison with M\ the
comparison of the two therefore does not lead to the
bare negation of their identity, but puts us on the track
of a supplementary b which must be added to <?, or an
alteration b which must be effected in a in order to make it
answer exactly to M. Now putting M down as equal
to a + b we make a second attempt and repeat the same
course of comparing and supplementing by fresh terms
c and </, till at last we get a definition M =. a + b + c + d
which in its expanded sum of characteristics exactly co-
incides with what we meant by M. In this very simple
process of thought rather than in a strictly inductive method
lay the art which the Platonic Socrates used ages ago to
clear up hazy conceptions.

CHAPTER II.

Of the limitation of Conceptions.

169. IN the course of an investigation we may be led
by a definite purpose to trace a group of characteristics
i k I through all the otherwise different objects in which
it occurs, and to ask what influence is exercised by its
presence upon the rest of their characteristics. The result
of this comparison then will itself teach us whether the
other characteristics which each of these subjects has in
virtue of the genus to which it belongs are modified by the
presence of / k I in any remarkable and particularly in any
constant manner. If this is the case we often form
out of / k I and out of the idea of a more or less pre-
cisely determined subject a new generic conception M^
treating all the ideas in which i k I occurs as species of
M. But 'whenever this is not the case (and not seldom
too when it is) we content ourselves with treating the
presence of //as one of the countless variable conditions,
which affect other ideas so far as to necessitate certain
alterations in them, but do not themselves form a generic
conception under which the several instances in which
they occur could be arranged as species. Now a living
language is believed by those who use it to have already
sufficiently distinguished in the coinage of its words the two
kinds of cases in which these two methods are severally
appropriate. Of course they will allow that enquiry, as
it goes deeper and deeper, will discover many a new group

LOGIC, VOL. I. Q

2?6 OF THE LIMITATION OF CONCEPTIONS. [Book II.

of characteristics ikl having such a decisive influence upon
the whole bearings of every conception that contains it as to
make it worth while to erect this group into a separate
generic conception M and to mark it by a name : and
language is in fact constantly enriching itself by new names
for the ideas thus newly discovered. But, on the other
hand, they will also assert that none of the conceptions
already found and fixed by the creation of a name are
unworthy of this distinction : each, they insist, really means
something coherent, which is thus justly cut off, as a whole
with well marked boundaries, from all other similarly co-
herent ideas.

17O. These conceptions which our inherited language
supplies are the tools with which our thought must work
and that not merely because we have no means of com-
munication except the words which have been invented
to express them : in this store of words is treasured up
the concentrated result of the thought which the human
mind has from the earliest times bestowed upon the world
to which it has access, and we may suppose that the same
impulses which led it to fix its conceptions in this form
would also in the first instance assert themselves in us were
we to go through the same labour.

But that these impulses, however natural they may be to
man, yet leave room for doubt is shown by the divergence
that constantly occurs in the application of the conceptions
thus formed. When the question arises whether some
predicate P is to be affirmed or denied of a subject S, one
maintains that is a kind of M and therefore is a P;
another objects that S is no M and therefore no P ; a third
allows that S is indeed no M 9 but an JV, but declares that
this does not matter, and that what is true of M holds
good of N also, while a fourth insists that the difference
between M&nd N establishes a difference between the two
in respect of P.

The divergence that here shows itself culminates in two

Chap. II.] DISPARATE SENSATIONS. 2^7

opposite tendencies, dominating the whole of our thought.
The one is a tendency to exaggerate every difference that
presents itself into absolute difference, and with the familiar
formula ' this is something quite different ' to resist all ar-
gument from one case a to another case b which resembles
a but is not exactly like it : this tendency becomes in life
and in science the spirit of the pedant and the philistine.
The other is a tendency to ignore the fact that a difference
which is not absolute difference may yet have a qualified
value, and with the barren phrase ' all is one at bottom '
to obliterate all the fixed boundaries which define the
province of each conception, thereby destroying the only
grounds upon which certain predicates are attached to
certain subjects and to no others : this becomes in thought
and action the principle of a no less ruinous libertinism.
A glance at the momentous consequences of these con-
fusions makes us alive to the necessity of clearly under-
standing what reasons there are to justify us in dividing
the whole extent of the intelligible world into definite
conceptions, where the boundaries of their several provinces
are to be drawn, and what value is to be assigned to this
demarcation.

171. We are led to very various issues by the attempt
to answer these questions even where they are easiest and
least pre n sing, viz. in regard to the simple contents of
sensuous impressions. We have a right to assume absolute
difference between simple sensations A B Cwhen we cannot
imagine any intermediate steps by which the peculiarity
of one could gradually pass over into that of another,
and when further we cannot think of any mixture of two of
them which would give a new simple sensation, and when
lastly there are no degrees of contrast between them such as
would enable us to estimate the difference between A and
B as greater or less than that between A and C or between
B and C. We find these relations, or rather this lack of
any assignable relation, between A B and C if A stand for

Q 2

228 OF THE LIMITA TION OF CONCEPTIONS. [Book II.

<

colour, B for sound, and C for smell. We may keep the
old name and call them disparate or incomparable.

This conclusion will not be affected by various secondary
considerations which may be urged. It may be pointed
out, for instance, that all three exist only as states of our
consciousness. To this we reply that they all are indeed
sensations, and may be called, according to the usage of
logic, species of sensation ; but that the conception of sen-
sation in general cannot serve here as a generic conception
in the sense of supplying a law for formation. When we
think of the shape of an obtuse-angled triangle as subordi-
nated to the general conception of a triangle, we have in
the latter a constructive formula, whose application has but
to be varied within its own limits in order to show us that
there are right-angled and acute triangles besides that one
species from which we started. But the subsumption of
colour under the general idea of sensation (for it is only
subsumption that is possible here, not subordination) can
never enable us to conclude from this general idea that
there are such sensations as sounds and smells besides
colours. Although these three then are, to use the ordi-
nary phrase, kinds of sensation, yet within the limits of
this universal they remain quite disparate the one from the
other.

Again as states, as motions or affections of the $oul, these
various kinds of sensation may produce certain secondary
effects that are comparable with one another, and it is
certainly allowable on that account to compare a certain
colour a 1 with a certain sound b* or a certain smell c l : but
still that which produces these comparable after-effects
remains itself quite incomparable. And we must make the
same reply to the physicist and the physiologist, when the
processes which must take place in the outer world or in
our nerves in order to produce the various kinds of sensa-
tion are traced back by them to comparable, or perhaps
even to closely allied movements of material particles :

Chap. II,] COMPARABLE SENSATIONS. 229

they must conclude not with the curious assertion that
there is therefore strictly speaking no qualitative difference
between these sensations, but rather with this other asser-
tion which is true, viz. that in spite of the similarity of
origin there is not the slightest similarity in the results.
There is no room for doubt here, except in so far as the
unprejudiced observation of ourselves, which is here the
sole criterion, is unable to pronounce decidedly. This is
the case with regard to taste and smell. Sourness is un-
doubtedly common to both; but the other sensations of
taste and smell also seem to form a connected group, only
that some members of this group are excited only by the
agency of liquids, others only by that of gaseous matter.
It may be that the sensations of these two senses, which on
this account must have different organs, are themselves
homogeneous and distinguished only by secondary sensa-
tions dependent upon the position, shape and action of
their respective organs. But it is not the business of logic
to decide this question : all we need do here is to warn the
reader when he has a direct perception that two modes of
consciousness are incomparable, never to allow this to be
overborne by sophistic arguments based upon the similarity
of their antecedents or consequents.

172. The other question, not as to our right to separate
A and ^,,-but as to our right to join together all that we
comprehend under A, calls for a similar remark. For a
long time people tried to dazzle the public with the stupid
paradox that black and white were no colours because they
did not like the prismatic colours depend upon a definite
number of undulations of light. The progress made of late
in the physiology of vision has completely cut away this
ground ; but even if this had not been done, no one could
have had the right to override language in this fashion.
Long before we knew anything about the exciting causes of
our sensations, language had invented the name of colour
for a group of sensations which by a homogeneous quality

230 OF THE L1MITA TION OF CONCEPTIONS. [Hook II.

<

directly perceived and undeniable, viz. by shining or what-
ever else we like to call it, are at once bound together and
separated from tones that ring or resound and scents that
are smelt Granted that the name shining is only appro-
priate to white and not to black, still the fact that the
fundamental quality thus imperfectly designated is shared
by both in common with the other colours admits only of a
verbal not of a real denial, and the common usage of the
term colour so as to include both was therefore completely
justified against the unsupported objections of the savants.
In other fields also we find similar instances of the en-
croachments of scientific theory, not always harmless in
their results. Thus chemistry for a long time contributed
to the confusion of speech by identifying oxidation and
burning. Men assuredly spoke of burning long before they
knew of oxygen, and always meant by it a process accom-
panied by visible light and sensible heat, which permanently
altered the constitution of a body : a glowing iron rod there-
fore was not said to burn, because no lasting alteration was
found in it when cooled : but also such a permanent change
would not have entitled the process which produced it to
the name of burning, in the absence of the sensible develop-
ment of flame and heat. The notion of burning then by no
means coincides with that of oxidation : many substances
are oxidized without burning, and on the other hand, when
heated antimony is immersed in gaseous chlorine and com-
bines with chlorine, throwing out flames the while, this
process is undoubtedly one of burning though not oxidation.
Geometers, again, knew ages ago that any system conceived
in abstract terms, i.e. arithmetically, provided that not more
than three scales be required for the arrangement of its
various elements, may be presented to our perceptions by
means of spatial constructions. Now there is nothing to
prevent a mathematician from c'onceiving systems based
upon any number of scales greater than three, only it is
plain that such systems can no longer be envisaged in space,

Chap. II.] TASTE AND COLOURS. 231

and that the name * dimensions ' which could be applied to
the scales in its ordinary sense of dimensions of space so
long as they were only three, can now bear only the more
abstract sense which I tried to express by calling them
scales. As space therefore means for us nothing but a
system that we envisage in this peculiar way which certainly
cannot be derived from any considerations of mere number,
to continue to speak of a system of four or five dimensions
as space is but to make sport of logical distinctions. Let
us be on our guard against all such attempts : they are
nothing but scientific freaks, which intimidate the popular
consciousness by utterly useless paradoxes and make it
doubt its well established rights in drawing the boundaries
of its conceptions.

173. When we now ask how the several coherent members
of one of those disparate kinds of content A B and C are
related to one another, we find that these relations are
peculiar and not always of the same kind. No one has yet
succeeded in reducing the several kinds of taste to a satis-
factory system : but the path which common usage takes in
naming them, incomplete though that nomenclature be,
seems to me the right path. Certain primary forms are
distinguished by names of their own, as sweet //, sour y,
bitter TT, and the others such as sour-sweet v ^ bitter-sweet
/x TT, are regarded as compounds of those well-marked
primary tastes. Our imagination could never have lit upon
this mode of naming them had it not been guided thereto
by the direct impressions of sense, for we cannot make
differences unless they are already present actually or
potentially in the data. Now these names imply that they
are actually present, not of course in the sense that the
sour-sweet is an aggregate of a sour and a sweet that can be
separated as much as if they were tasted at different times,
but in the sense in which we speak of a mixture as opposed
to an aggregate. The fact that such a mixture is possible
here, i.e. that sour and sweet may be united in one im-

232 OF THE LIMITA 7ION OF CONCEPTIONS. [Book II,

pression in a manner that we can scarcely describe but
easily feel, while sweet and red cannot, distinguishes the
relation of the several tastes to one another from that of the
disparate groups ABC.

It may be objected that in the sour-sweet the difference
between the sour and the sweet is only present potentially
not actually; that there may easily be a third impression o>,
itself simple and in no way compound, yet forming a con-
necting link between /x and v, and that this then, on account
of its resemblance to both, is designated in speech by the
two limits /x and v between which it falls, without implying
that it actually is a mixture of the two. This objection
I should not consider sound unless there were present in o>
besides that in which it resembles /x and v an independent
remainder that could not be accounted for by the combina-
tion of /x and v ; where this is not the case this third im-
pression o> will not merely be called a mixture /x v by an
arbitrary freak of fancy, but will in fact be that and nothing
else. But the primary forms /x v TT and all mixtures of these,
though made one group by the fact that they all alike
appeal to the sense of taste C, yet within those limits can
only be regarded as disparate from one another. A man
who had tasted nothing but sweet could never by any con-
ceivable modification of the feeling it gave him discover
the peculiar nature of sour or bitter that he had not yet
experienced. There is then no transition from /x "to v or TT
through independent connecting links, but we must first
know fx v and TT and then get the intermediate links by
various mixtures of these.

We find the same relations between colours, and I took
occasion in an earlier passage 1 to justify the common usage
of speech in always distinguishing a limited number of
primary colours, and inserting the rest as mixed colours
between them. It is of course possible to lead the eye
gradually through skilfully selected middle-tints from the
1 [Sect. 13.]

Chap, II.] THE SCALE OF SOUNDS. 233

impression of one colour to that of another : but while red
passes into orange or violet only by an admixture of yellow
or blue which can still be felt as yellow or blue, that which
makes red what it is does not pass over into that which
makes blue what it is. A man who had experienced one but
not the other could never discover in the simple nature of
red anything which could possibly be modified, heightened,
or cooled down in such a way as to lead him to imagine
what blue is : he would have to learn what blue is before he
could mix the two extremes together so as to arrive at the
intermediate violet. The modifications of which the several
primary colours are capable must also be regarded in the
same way. We undoubtedly have the right to consider
bright blue and dark blue as kinds of the same bJue : but
these kinds also are produced by the mixture of white or
black with a pure blue that is always the same though never
visible in its purity. Only I would once more briefly re-
mind the reader that all that I have hitherto said refers only
to the nature of our sensations after they have arisen in our
consciousness, and has nothing to do with the physical or
psychical conditions of the act of sensation.

174. With sounds the case is essentially different. After
a comparison of several sounds we distinguish first of all
three predicates. The peculiar tone of the instrument
which is sounding, whatever the physical antecedents may
be, is for our feeling a simple property which defies further
analysis, more analogous to a taste than to anything else.
However strongly we may be moved by the secondary
effects of this peculiar tone the essential nature of the note
seems to us to be quite independent of this, and also of its
second property, viz. its loudness or strength : we regard
both only as ways of producing the same note, the dis-
tinctive nature of which lies in its pitch. But in this third
aspect sounds do not like colours fall into a number of
distinct stages, such that one can pass into another only by
mixture, but they rather form a continuous series, in which

234 OF THE LIMITA TION OF CONCEPTIONS. [Book n.

the difference between two more distant members is only
a multiplication of the difference between two adjacent
members. It is impossible to make a proportion in which
red shall stand to blue as yellow to any fourth colour : but
the difference between two notes can always be stated as a
multiple of some difference which we take as the unit.
This difference itself is of a quite peculiar kind : we should
not use the phrase * higher ' and ' lower ' in speaking of
sounds, unless, quite apart from the frequency of the sound-
waves which we certainly do not feel, our feelings themselves
announced one note as a heightening of another : but this
quantitative idea cannot be referred here as it can elsewhere
to a qualitative content that is independent of it : a note d
is different from another c even in quality just because in
it the undefmable common property of sounding which it
hhares with c is c heightened ' in that peculiar way which we
can only express by this happy metaphor, or at most by the
more technical phrase * qualitative intensity.' The differ-
ences of notes therefore are homogeneous and measurable
in extent, which the differences of colours were not : the
notes intermediate between two others arc not formed by
mixing these two together, but are on a footing of perfect
equality, as original members of the series, with those
members between which we place them.

And lastly the whole scries is endless : it is not possible,
in addition to the colours known by experience, td imagine
a new colour of which we can have an idea though it
happens that our eyes never saw it ; the scale of sounds, on
the contrary, may be continued ad infinitum because each
is generated out of its predecessor by a heightening which
is felt to be homogeneous. It is not unmeaning to talk
of sounds higher or lower than any that can ever come
within our experience, because we have here (what we could
not have if we tried to imagine new colours) a distinct idea
of the way in which these sounds would differentiate them-
selves 2/"they were audible.

Chap. llj THE SENSATION OF HEAT. 235

175. With some modifications, which I leave the reader
to make, these remarks apply also to the series of our
sensations of heat : but at the same time the latter exhibit
a new feature. The living body's own need for warmth
gives a peculiar significance to certain sections of the
series ; we distinguish cold, cool, lukewarm, warm, hot, and
fancy that these terms have a definite meaning; but not
only would it be impossible to draw a hard and fast line
where cool ends for everybody and lukewarm begins, but
even if we interrogate our own feelings merely we are
obliged to confess that there must be a certain caprice in
choosing the one name or the other. We may connect with
this contrast of heat and cold, and of high and low sounds,
a great number of other pairs of ideas, the content of which
is not so directly derived from sensations, e.g. great and
small, strong and weak, many and few, old and young, and
many more of the same sort.

However decided a contrast is intended by the two terms
of these antitheses it is always impossible to mark off the
province of the one from the other, they constantly and
insensibly pass into one another. But when we go through
such a scries the passage from a to z and that from z to a
are very clearly different, -to some extent they admit of
definition, and our immediate feeling at any rate never fails
to distinguish them. We cannot say what is warm nor
what is cold, but we can say without any doubt whether a
is warmer or colder than b : in this case the decision is
a matter of sensation ; in passing from a to b we are con-
scious of a change which is the opposite of that which we
experience in passing from b to a. We cannot say what
great and small mean, but the statement that a is greater
than b is quite free from ambiguity, and may be defined to
mean that if b is taken from a there is left a positive re-
mainder <5. And it is the same with the other examples :
these adjectives are all derived not from the apprehension
of one idea but from the comparison of several, and denote

236 OF THE LIMITA TION OF CONCEPTIONS. [Book II.

relations which have no fixed value or meaning apart from
a second point of comparison. These adjectives therefore
are indefinite in the positive ; only their comparatives have
an unambiguous meaning. Where the positive form is used
in speech it means that the comparative term may be applied
to the thing denoted when compared with an unexpressed
standard, which either in the estimation of the speaker or
in common opinion is the normal or usual state of the thing
in question.

176. There is one more point to consider in connexion
with sound and sensations of heat. Sounds being in them-
selves of perfectly equal value we have no inducement to
select some few of them as fixed points and to give them
prominence by naming them. But on aesthetic grounds
we want to articulate the whole series. As the simple sen-
sation of a note is undefmable we characterise it by stating
the cause which will at any moment produce precisely that
note, i.e. the frequency of the vibrations upon which it de-
pends. But there is no reason for preferring one number
to another, and as every member of the series may be
defined with equal ease in the way named, the musical scale
has in fact no absolute starting-point. It is true that other
circuinstances, viz. the harmonic relations of notes, which
I must here pass over in spite of the interest which they
have even for the logician, lead us to arrange the series
in octaves ; but even this arrangement has no fixed starting-
point ; we may begin at any height we please.

Our sensations of heat do not admit of such a simple
definition by their causes ; we are obliged to have recourse
to the other observable effects of their unknown cause, viz.
the expansion and contraction of bodies. To take the
melting-point of ice as the point from which the degrees of
temperature should be measured in an ascending and
descending scale was to choose a quite arbitrary zero to
reckon from, though one very well adapted to its purpose :
for the fluidity or solidity of water is a point of cardinal

Chap. II.] ARBITRAR Y STARTING-POINTS. 237

importance in the meteoric and organic processes which
surround us. But it is after all merely a zero in our cal-
culation, not in the thing calculated. Starting from the
unknown amount of heat (call it x) which is present at the
melting-point of ice, all we do is to reckon the increase or
diminution of this amount by multiples of a unit-degree
chosen expressly for this purpose. Thus 12 is not the
double of 6, but the difference between o (which is equal
to x) and 12 (which is equal to x -f 12 units) is twice as
great as the difference between o or x and 6 (which is
equal to x -f 6 units).

The reader may see by this simple illustration that though
a series or a complex system cannot be articulated and ar-
ranged in a regular order unless there be a corresponding
regularity in its own relations, yet thought frequently has to
take a quite arbitrary starting-point and an arbitrary stan-
dard in order to master and make use of this regularity;
and that such an arbitrary arrangement, though admitted by
the nature of the object and justified in its results, yet must
not be looked upon as a property inherent in the object
itself.

177. Practical life offers many illustrations of this remark.
We here have to do with qualities which either attach to
various persons and things in very varying degrees, or which
in one an.d the same subject take successively a continuous
series of values, from which proportionate effects are ex-
pected. But it is only in nature that effects vary con-
tinuously in accordance with the conditions : where the
result does not follow till it is produced by human action,
the exact observance of the desired proportion is generally
prevented by the fact that the labour required would be out
of all relation to the end in view. We have to content
ourselves with breaking up the whole series of values into
sections and acting as if the conditions were the same
throughout each section, fixing the result at an average
amount, which will be too great for the first and too small

238 OF THE LIMITATION OF CONCEPTIONS. [Book II

c

for the last members of that section of the series. Thus for
the purposes of taxation we divide the series of properties,
from absolute poverty up to the highest pitch of wealth that
is likely to be found, into a number of classes; in calculating
the premium to be paid on a life-insurance we reckon age
by years or at lowest by some considerable fraction of a
year ; in calculating interest we keep to a day as an in-
divisible unit. Again it may happen that a quality gradually
attains a certain pitch to the attainment of which we desire
to attach certain consequences, though we cannot say at
what moment the decisive condition is fulfilled. That
maturity of body and mind which we have in our minds
when we say that a man is of full age or has attained his
majority is certainly attained by different persons at different
times of life ; but it is impossible to find out the actual
moment in each individual case, not merely because it
would necessarily be an endless business to appraise the
total merit of the person, nor yet because such a censorious
proceeding would be unjustifiable, but because, though the
higher grades of maturity and immaturity are easily recog-
nisable, there is really no certain mark to distinguish them
in doubtful cases. But for all that the needs of social life
require that a definite time be fixed ; so the law has to fix
it summarily, and attaches to the completion of certain days
and hours the beginning of certain rights and duties, though
no one supposes that the capacity and the obligation which
were absent yesterday have actually sprung up in the course
of the night. But though this proceeding is summary it is
not without reason : the choice is limited to times which
correspond without any appreciable difference in accuracy
to the requirements of the situation ; all that is arbitrary is
the preference of one out of a number that would all do
equally well.

There are other cases in which we are still further from
finding any precise standard in the nature of that which has
to be settled, and must look for it in the further ends

Chap. H.) DEGREES OF GOODNESS. 239

whose attainment is to be facilitated by the settlement.
Such are the fixed periods within which certain conditions
must be satisfied in order to establish some legal claim or
to avoid some legal obligation : though the outlines of these
arrangements are determined by the object specified, their
details aim at nothing but logical precision. This our
ancestors effected by not measuring the more important
periods by entire units of time of the larger kind, but
adding to such units some fraction of them, some days to
a week, some hours to a day; by these means they nar-
rowed the period within which (to use a common but
rather loose phrase) a man might have fancied that he
was satisfying the law. The police again are quite right
when in order to prevent disturbance of the peace they
summarily fix the number of persons that shall be held
to constitute a forbidden assembly at three or five, thereby
barring disputes like those the old sophists used to raise
when they asked how many grains of corn are required
to make a heap, or how many hairs must be lost to make
a bald-head.

178. To return from this digression : -whether a note is
to be called high or low, a liquid hot or cold, are questions
that people never quarrel about : there are no interests
attaching to the content of these conceptions that could
make us -hesitate to admit at once that their meaning is, as
we said, relative. It is different with good and bad. We
set the highest value on the fixity and absoluteness of
these conceptions : every action, not simply as compared
with others but as it is in itself, must it is thought be
unequivocally included in the one and excluded from the
other ; people even think they are bound to deny that
there are any degrees of goodness in the good or of
badness in the bad, for fear lest the diminishing values of
the two should at last meet in the indifferent as a zero-
point, and a constant transition be thus set up between
two opposites which ought rather to be severed by breaking

2 40 OF THE LIMIT A TION OF CONCEPTIONS. [Book II.

r

down every bridge. But this logical rigour is utterly at
variance with the unprejudiced judgment which we all bow
to in real life. No one really doubts that there are degrees
of goodness and badness, and no one can persuade us that
no acts are indifferent till he has artificially limited the
conception of an act. But it really is no use to try to fend
off the threatened confusion of good and bad by first
dividing all actions into those which can be judged morally
and those which cannot, and then proceeding confidently
to divide the former into two absolutely opposed groups,
the good and the bad. We thereby do but move our
doubts a step further back ; for the question now is where
is the line to be drawn between that which calls for a
moral judgment and that which does not ; and this line
as before will seem to vanish in a perpetual passing of the
one into the other.

Again the relation of the pleasant to the beautiful and
the good, though a less pressing question, is one of great
interest on aesthetic grounds. To the man without a
theory they seem to arrange themselves in an ascending
series, not merely according to their value but according
to the meaning of their content ; not of course in the sense
that by mere intensification what is extremely pleasant
would become beautiful, or the highest beauty pass into
the lowest grade of goodness, but in the sense that there
are kinds of the pleasant, distinct in quality, which begin to
have a right to the name of beautiful, and forms of beauty
which produce an aesthetic impression akin to moral ap-
probation. But those who theorise upon morality and
upon art alike resist this admission ; they deem the beauti-
ful falsified if it has anything to do with the good, the good
degraded if it has anything in common with the beautiful,
and through this with the pleasant. Here too, with regard
to beauty at least, people have been found to deny all
differences of degree, and to maintain that what is beautiful
at all is entirely beautiful, and that if you allow there is

Chap. II.] TRANSITIONS. 241

anything more beautiful, you cannot think this really beau-
tiful at all.

179. Let us, in order to settle these doubts, look around
for other illustrations. Of the straight line, from its nature,
there is of course but one species known to the geometer ;
but in curves he distinguishes countless degrees of curva-
ture of measurable value, so much so that the straight line
itself appears as the extreme limit to which the curve con-
stantly approximates as the radius increases. Yet in spite
of this unbroken continuity not merely does the geometer
persist in the general statement that curved and straight
are opposites that can never be reconciled, but no doubt
ever arises in its application to a particular line which is
accurately known ; however near it may come to a straight
line it is yet quite undeniably curved, so long as the radius
of curvature has any finite magnitude.

Again a curve may in one portion of its course be con-
cave to an axis to which it is convex in a further portion ;
if it makes this change of direction in an uninterrupted
sweep without any angle that breaks the continuity, there
is no doubt that its tangent at the turning-point, and there-
fore the element of the line itself, is parallel to the axis in
question, and so neither concave nor convex ; but although
both directions thus visibly meet in one zero point of
indifference that belongs to neither, yet the opposition
between them is thereby neither altered nor removed ; on
this side of that point the curve remains entirely concave,
on that side entirely convex. Take a simple instance :
between i and 2 we may insert countless fractions rising
gradually in value from i to 2 ; between full daylight and
midnight darkness countless degrees of illumination not
only are conceivable but actually occur ; between pleasure
and pain there lies an uninterrupted series of feelings which
connect the one with the other : but i does not on that
account become equal to 2, nor do darkness and pain
cease to form a perfect contrast to light and pleasure ; and

LOGIC, VOL. I. R

2 ^ 2 OF THE LI MIT A TION OF CONCEPTIONS. [Book II.

at the same time each member of these pairs, by itself and
without reference to the other member, is something so
definite that we never mistake the one for the other. These
illustrations are sufficient to explain the statement that the
existence of countless degrees through which two opposites
A and B pass till they meet in a common zero-point of
indifference, does not destroy the difference or opposition
between the meanings of A and B themselves.

180. And so even if the moral philosophers had succeeded
(and it is their business and not ours at present) in deter-
mining what they mean by good A and bad B as precisely
as the geometer defines what he means by convex and
concave, they would still have had no ground for denying
that good and bad have degrees and meet in the indifferent,
in order to maintain unimpaired the distinction between
the two. The specific meanings of the general conceptions
good and bad are not in the least degree altered because
particular cases to which the terms are applied partake
more or less fully of the character of one or the other of
these opposites. But the zero-point of indifference can still
less contribute to the confusion of the two, for its meaning
is not that both are true at this point, but that neither is
true ; it is therefore merely a point of separation, on this
side is only good, on that side only bad.

On the one hand then the maintenance of the distinction
between good and bad is no reason why people should
deny that there are degrees of good and bad ; on the other
hand we must insist upon an explicit admission of the fact
that there are degrees. To deny it, to repeat the old Stoic
paradox omnia peccata esse aequalia, or to go on preaching
that even the smallest error is still not truth but error and
nothing else, is but to waste time in tedious assertions which
as they contain only half-truths may on this very principle
be called errors and nothing else. It is not true that ,a
curve is once for all a curve, so that the degree of its con-
vexity or concavity is quite a secondary consideration,

Chap. II.] AMBIGUOUS CASES. 243

which has nothing to do with its character as a curve ; the
fact is that one curved line is actually more curved than
another, and so realises more intensely the character
common to both. Similarly the good or bad intention
out of which an action springs can not only be measured
in a secondary way by the importance of the interests
affected by the act or of the circumstances under which it
is done, but can itself be estimated according to its degree
of goodness or badness; for such an intention is by no
means a mere form which is alike in all cases ; it is an
inner process which not only must reach a certain degree
of intensity in order to generate the impulse which every
act requires or to overcome certain obstacles, but has also
a certain degree of value according to the amount of the
good or evil which it consciously aims at producing. Error
again is not merely not-truth ; that would not distinguish it
from doubt ; it is a departure from truth, and has therefore
a measurable magnitude, indeed is inconceivable without
it ; a man whose thoughts are occupied with real problems
therefore will not be so silly as to reject in identical terms,
as mere errors, two assumptions of which the one is so far
from the truth that it leads to no knowledge at all, and the
other so near that it leads to nearly all the knowledge of
the subject that can be expected.

181. It may be that the series of the pleasant, the beauti-
ful, and the good (the further consideration of which I
leave to the reader) has already suggested another relation
that can exist between a series of conceptions, which I will
first of all illustrate from geometry. Imagine two pyramids
A and B presenting similar horizontal sections but one
sloping more steeply than the other ; if we place them so
that the apex of the one (the less steep) lies within the
other (the steeper) and upon. 1 a point in its axis, then the
plane which passes through the intersection of their surfaces
belongs both to the series of planes of which A is the
integral and to the series of other planes the endless succes-

R 2

2 4,4 OF THE LI MIT A TION OF CONCEPTIONS. [Book II.

sion of which is summed up in B : similarly we can imagine
a third pyramid C which should in like manner have a
plane in common with B.

Now the generating law of each of these solids, with
reference to the common axis of all three and the position
of the apex in that axis, may be stated in a formula, which
would have to be compared with the general conception of
A B and C respectively. It would then appear that in the
A series there is one member that also satisfies the require-
ments of B \ and therefore that as to this member it is a
matter of doubt or of indifference whether it is to be classed
under the conception A or .Z?, not because it satisfies
neither, but because it perfectly satisfies both at once. But
with the exception of this particular case all the instances
of A, all the other planes by which the compound solid thus
formed could be intersected, would belong exclusively
either to A or to B. The same would be true of the plane
common to B and C.

In these cases then it is due to the very nature of the
essentially distinct conceptions that certain members of the
series which they severally characterise become ambiguous,
so that by themselves and without taking count of some
secondary point, such as the manner of their origin and
development, it is not safe to ascribe them exclusively to
any one of these conceptions, though apart frm these
particular cases there is no doubt at all about the difference
of the conceptions. We have here named ABC and so
expressed them as conceptions, leaving the particular cases
unnamed. But the purposes of speech may sometimes
suggest the opposite procedure. We may name and fix
certain conceptions M N O which have quite unambiguous
and distinct meanings only in particular cases, which we
may picture to ourselves as salient points, as maxima or
minima, in a connected series. We shall then find the
reverse of what we found just now, i. e. we shall find many
contents furnished by feeling and experience which have a

Chap. II.] CASES IMPERFECTLY CLASSED. 245

3

place indeed between two of these conceptions, but only
between them, corresponding completely to neither.

182. As illustrations of the latter procedure we may take
compound conceptions got by starting not from one but
from many points of comparison at once. With such a
conception no doubt every instance agrees which in each of
these respects is found to have the appropriate mark ; but
the applicability of the conception becomes doubtful in
many other cases, which from one point of view would
certainly be included under it, but from another which
must also be considered would certainly not. Various
thoughts thus cross one another in the conception of illness.
Illness is certainly above all things a departure of the bodily
condition from a supposed fixed standard. But a mal-
formation, which departs considerably from the natural
structure of the body, still cannot be called an illness, so
long as it does not impair the vital functions, nor so long
as it remains constant and runs no natural course through
various stages. A wound always in some degree alters
structure and function, and also runs a natural course ; but
a slight wound is not called an illness, plainly because it
does not involve danger nor make the body unserviceable
for any important purposes of life ; but again a very severe
wound is also not called an illness though it does both ; its
origin is, too sudden and too entirely due to external
violence, and now we observe that when we spoke of
illness, we thought of a state which, though dependent
upon some external cause for its origin, yet takes its definite
shape from the peculiar interaction of the internal forces.
But now a cold is such a reaction of the internal forces against
an external stimulus : but a cold is scarcely called an illness
so long as the element of danger is absent : and just as we
here help ourselves out with the milder phrase f unwell/ so
we use the term health with a certain latitude, allowing
room for the slow advance of a number of disturbances
connected with individual idiosyncrasies.

246 OF THE LIMITA TION OF CONCEPTIONS. [Book II.

It is not difficult to say what is the right course here. It
is impossible in such cases to find a definition which shall
be in harmony at once with the requirements of science
and with these strange caprices of language : if we want to
determine the conception, we must disregard usage and fix
it arbitrarily. In the instance we have chosen this is* scarcely
needed, for pathology gets on very well without any unim-
peachable definition of the nature of illness in general ; and
the physician has absolutely no need for logical generalities
which yield no guidance in practice.

But in other cases it is not so. In our conception of
crime all sorts of consideration cross one another, we con-
sider whether it was deliberate or precipitate, what was the
degree of evil intention, whether it was attempted only or
perpetrated, what was the amount of harm done : the dis-
tinction between the creations of art and the products of
manufacture, or the relation of a free reproduction to a
literal copy, presents similar ambiguities. To fix the limits
of the conceptions is of more importance here, since by the
operation of law certain advantages and disadvantages
follow regularly and directly according as a given case is
judged to belong to the one or the other ; but here also,
though we take count of common usage, it is yet necessary
in the main to distinguish them by positive enactment.

183. Obviously we may set down any conception M as
equivalent to any other conception N when we have by
further specification so changed TV that it is equal to M.
Thus there arise a number of incidental aspects or va-
riations of the expression for the same M, which we shall
further on find to be of use in enabling M to be subsumed
now under this law and now under that, such law leading
to a new assertion about M. There is no limit to the
extent to which this procedure may be legitimately carried
so long as the transformed M really coincides with the
original Jlf 9 so long that is as TV is equal to M. We
may even bring a triangle M under the conception of a

Chap. II.] TRANSFORMATION OF CONCEPTIONS. 247

four-sided figure N^ provided of course that we add triat
one of the four sides is reduced to nothing. This may
seem mere trifling, but it is useful in practice : we can
thus for instance easily picture to ourselves how every
time that two sides of a polygon, which were before sepa-
rated by an intervening side, are made to meet at their
extremities by the vanishing of the intervening side, the
sum of the angles of the polygon (in this case four-sided)
is diminished by two right angles.

This use of transformation will engage our attention
further on; what I wish here to emphasise is that the
difference between the two conceptions thus brought to-
gether is of course not altered by it. The four-sided
figure remains just as distinct from the triangle as it ever
was, i. e. so distinct that it must be stripped of its very
essence before it can be ranked with the other; and
similarly the alterations, whatever they be, that must be
made in order to turn N into M, give the measure of
the abiding difference between the two conceptions. When
we are dealing, not as in this case with abstract con-
structions of thought but with realities, which have an
independent origin in the region of fact, such transfor-
mations have very little value ; they are in the first instance
mere fancies, whose significance cannot be ascertained
without special enquiry. In thought we may change any
given form of crystal into any other that we please by
cutting off slices here and there, by successive alterations
of outline we may change the likeness of a crocodile into
that of a bird, from any one chemical element we may
in thought derive all the others by giving successively
certain other values to the coefficients which the funda-
mental properties of matter take in the case of that one.
But by such devices we cannot make the conceptions
M and N approximate to one another, for their difference
remains always as great as the number of steps that we
must take to get from one to the other; neither can we

248 OF THE LIMITA TION OF CONCEPTIONS.

thus establish between the actual things which exemplify
these conceptions such a connexion that one might pass
over into the other. For that it would be necessary to
prove that the physical forces of the elements which build
up an actual crystal of the form M are such as to make
it possible for the same elements to be also in equilibrium
when arranged in the form N \ or -that the concatenated
system of forces which determines the structural type of
the crocodile and maintains it in life may be so modified
by other natural influences that the form of a bird may
actually grow out of it, that in short the order of nature
actually contains impulses which realise the changes which
we may choose arbitrarily to make in thought or upon
paper. We cannot but remember, though happily as an
error which we have outgrown, the wild caprice with which
not long ago people would derive a word in one language
from any casual word in another, and call it etymology;
at the present day people need to be warned against pro-
ceeding in a similar way to satisfy the newly-awakened
desire to conceive all the various kinds of organic beings
as evolved from one other, all fixed specific differences
being done away. But, whether Darwin has succeeded
or not in his attempt, we must at any rate allow that he
has taken the greatest pains to point out the real processes
of nature by which the transformation of one organic form
into another which we can conceive in thought may have
been actually brought about.

CHAPTER III.

Schemes and Symbols.

184. IN this chapter I shall continue to treat of the same
subject as in the foregoing, but from a somewhat altered
point of view. The extent and importance of the difference
between several ideal contents can, we ascertained, be
precisely determined only when we find ourselves able to
compare several differences of the same kind, i. e. when the
ideas to be compared themselves form series, whose
members proceed according to a law that can be more or
less exactly stated, and when moreover from the nature of
the feeling whose modifications, distinct both in quantity
and quality, are represented by the members of the series,
such modification can only take place in one and the same
direction. Compound conceptions whether of things or
properties^ situations or events, by reason of the number of
the characteristics or of the aspects which they include,
may be altered in various directions ; one or some or all of
these characteristics and of these aspects may run through
all the various phases of which they are capable ; and again
the bonds which connect them may pass through all the
various degrees of laxity and strictness and all the changes
of form to which they are by their nature liable.

Now there is no reason why the value or the extent of
the difference between two such compound conceptions M
and N should not frequently be revealed to us by a direct
impression with as much certainty as we need require in

250 SCHEMES AND SYMBOLS. [Book II.

t

the case in question : if however a more accurate determina-
tion were needed for scientific purposes, we should have
first to determine the values of the various scales upon
which the several alterations take place, and thence to
determine the value of the total alteration which separates
jft/from jfVor jA^from O. The reader may be inclined to
object at once that in most cases at any rate we proceed in
the reverse order to estimate the significance of the scale of
a change which has taken place by the amount of the
change which this alteration has produced in the total im-
pression. I may allow this objection without taking any
further notice of it ; for what I here wish to illustrate is not
a logical rule but a propensity of our reason, which needs
to be checked rather than to be indulged, but which as it is
ineradicable needs to be specially mentioned. It is easy
to understand, I mean, how out of the above-mentioned
problem may arise the wish to have a universal scheme in
which not only all the modifiable relations of different
elements that we can think of, but also the values of
the difference between any two modifications should be laid
down so completely that the difference or the kinship
between any two conceptions Mand ^should be exactly
indicated by their position in the universal scheme.

185. To illustrate this I will first go back to remote
antiquity, to Pythagoras, To reconstruct a body of genuine
Pythagorean philosophy out of the scanty and for the most
part very questionable materials at our command is a task
which I will not undertake, but I think I am able to state
what may have been the fundamental idea which animated
it, and which would enable us to understand why the
sympathy stirred by it has been so lasting though often so
perversely expressed. It is tolerably certain that the bent
of the school was first to abstract mathematics, and
secondly to their application to the processes of nature.
The first line of study could not fail to lead them to picture
the series of numbers and the world of shapes as two great

Chap. III.] PYTHAGOREAN SCHEME. 251

coherent systems, and further to bring them to see how
spatial figures themselves depend upon the numerical
magnitudes which they involve. The second, besides other
less known results, led to the discovery of the relation
between the pitch of a note and the length of the vibrating
string, and thereby no doubt suggested the general idea
that even phenomena whose differences are in the first
instance felt by us as differences of quality are based upon
mathematical differences that admit of comparison. The
rash generalisation of results thus won is what the fancy of
men is always prone to ; the mathematically-trained Pytha-
gorean went so far as to make the reflexion that if it be
once established that a scries of changes in phenomena
corresponds to a series of changes in magnitude, then every
other conceivable mathematical relation along with all its
modifications must have its counterpart in the phenomena,
or conversely, if a group of phenomena is based upon
definite relations of magnitude, the coherence of all the
processes of nature necessitates the conclusion that all other
phenomena also depend in like manner upon relations that
can be mathematically determined.

This I conceive to have been the origin of those specula-
tions which Aristotle expresses by saying that Pythagoras
regarded the principles of numbers as the principles of
things : but we must further consider the meaning of this
expression. The purport of the Pythagorean philosophy
was certainly wider than we might be led to suppose by
that other saying of its author, that God has ordered every-
thing by measure and number; i.e. it was not limited to the
mere application of mathematics to nature, if that means
merely that the definite magnitudes of natural forces and
processes modify one another when brought into contact
according to the same mathematical laws that hold good
for magnitudes in general : these data themselves, to which
mathematics are merely applied by modern * mathematical
physics/ were regarded by Pythagoras as in themselves form-

252 SCHEMES AND SYMBOLS. [Book II.

ing a system whose inner articulation is based upon the same
relations that determine the structure of the series of
numbers and of all their possible combinations. I wish to
distinguish in this theory a general idea and the particular
form given to it.

186. The so-called natural philosophy of the lonians had
devoted itself to describing the processes by which natural
bodies were formed out of their primitive matter and
returned to it again/ As this philosophy very generally
used for this purpose the ideas of condensation and
rarefaction, it may appear, in virtue of its employment of
quantitatively determined means, to be closely akin to the
Pythagorean theory. The two are nevertheless very far
apart : for the lonians never betray any desire to show that
the sum of that which is thus produced at any moment of
its existence or in the whole series of steps by which it
comes into existence forms a coherent whole of mutually
dependent parts. Pythagoras on the other hand seems to
have troubled himself very little about this origin of the
world, but the world as it was after it had come into
existence was to him a system, such that not merely were
its parts there, one beside the other, but that there would
have been a gap in it if while one phenomenon were
present another had been absent. If a and b and d are
present, then if c is there at all, it is not merely there along
with the others, but it is there because the law according
to which the series a b advances to d requires it as the third
member of the series which is indispensable to the presence
of the fourth member d : or if c is absent, it is not merely
absent as a matter of fact, but because the law which
regulates the series excludes the possibility of this third
member before d. The same consideration may be applied
to other series in the actual world, to a /3 y 6 and to a fc c ft,
and this application was made by the Pythagorean school.

How they conceived the relation between the different
characters of these series, which I wished to indicate by the

Chap. III.] A SERIES OF TYPES. 253

*

use of different alphabets, is a point upon which we are
certainly in the dark, and upon which, as we may gather
from Aristotle, the fullest information would probably throw
but little light ; but with respect to the law which in each of
these series binds the homogeneous members together,
it seems to be indubitable that it was regarded as precisely
identical for all the series, i.e. that they maintained a
complete parallelism between the relations prevailing in the
various groups of connected phenomena. This is shown in
the supposition that the earth has an invisible fellow, in
order to bring the total of the then known planets up to ten,
to which number the arithmetical mysticism of the system
had once for all assigned a peculiar significance, in the
assumption of a fifth element, which together with water,
earth, fire, and air, shall correspond to the five regular solids,
tetrahedron, cube, octahedron, dodecahedron, eicosahedron,
in the attempt again to conceive the distances of the
planets as arranged according to musical intervals, and
even in the meagre form of their tables of opposites. To
us of course these tables do but illustrate the frequent
occurrence of this relation of opposition between two con-
ceptions even when these are arbitrarily chosen, but the fact
that they always contain ten pairs seems to indicate that
they were intended to represent this relation as essential
for all ttje different stages in a series of ten members.
Finally when they assigned life to the number six, intelli-
gence and light to seven, and friendship to eight, we see
that they regarded not merely the phenomena of nature,
but also those of mind, and in a word every conceivable
thing, as ordered according to the same serial law.

This philosophy then sought and fancied that it found
precisely what we spoke of above, viz. a universal scheme
which mounting from simple to complex was supposed to
embrace the whole sum of possible forms, one of which was
to serve as a pattern for the formation of every actual thing,
while at the same time these forms or types were to be so

254 SCHEMES AND SYMBOLS. [Book II,

arranged in the scheme that the position of its type directly
determined the significance of every actual thing, and the
amount of the difference or the kinship between it and
other things formed upon the model of other members of
the series. The general idea then that I would ascribe to
the Pythagorean philosophy is this, viz. not merely a subse-
quent arrangement of things whose nature was originally
settled without reference to the principle of this arrangement,
but a harmony of the Cosmos which name was first applied
to the world by Pythagoras based upon the notion that all
things are from the beginning nothing but various realisa-
tions of a series of types, regulated by one law of development
which is the same for all.

187, The general conception is undeniably grand, but
grandeur is sadly lacking in the special form here given to
it. Even in the present state of the mathematical sciences,
various as are the magnitudes whose interesting mutual
relations have been examined, it would be impossible to
find adequate types or symbols or abstract expressions for
the still more various relations that subsist between the
elements of the actual world and the combinations that
arise out of them ; but the arithmetic of the ancients, which
the Pythagorean school seems to have helped to develop,
furnished in its then state but very few and very meagre
numerical relations, whose significance must have been
much exaggerated and from the beginning very arbitrarily
interpreted before they could be regarded as the relations
upon which the structure of the world is based. The
grounds on which they justified their well-known veneration
for the number ten, viz. the fact that all numbers are
generated by the repetition of unity; that in this series
the even numbers alternate with the odd numbers, which
cannot be divided by the principle of multiplicity,' i.e. by
two, and which are therefore held to be of higher rank;
that three is the first union of odd and even, four the first
square of a multiple number, and ten the sum of these

Chap. III.] THE 'PRINCIPLES' OF NUMBERS. 255

exalted four first numbers, are grounds which could not
be admitted except by a system of symbolism which was
ready to accept any interesting motive without regard to
its connexion with others : though the real grounds of that
veneration undoubtedly lay in the habitual use of the
decimal system. If these thinkers had been acquainted
with all the algebraical and transcendent forms of functions
which are the instruments of modern mathematicians, how
much more various would have been the symbols employed,
and how much more delicately would they have been
adapted to the nature of the several phenomena ! The
same tendency still survives in us : even in cases where
calculation in the strict sense is impossible we are inclined
to use the term 'power' 1 when the meaning and importance
of a conception is raised in some peculiar manner, as for
instance when each of the centres of relation, whose deter-
mination by each other constitutes the meaning of the
conception, is itself exalted into a small system, whose
members determine each other in the same way.

We can imagine then how the Pythagoreans (if they had
had our knowledge) might have illustrated many relations
of dependence between various elements by the relation
of a logarithm to its number, and how they might have
applied trigonometrical functions to explain any kind of
periodicity. As however they had not our resources at
command, and as even these would still be insufficient, it
would be quite useless to examine in detail the reasonable-
ness of the Pythagorean symbols.

188. That it was the fate of the whole theory to be
variously interpreted and misunderstood is easily explained
by its nature. According to one statement of Aristotle it
was the principles of numbers that Pythagoras identified
with the principles of things. This seems quite intelligible.
By these principles of numbers must be meant the relations
between one and the other numbers, the way in which one
1 [In the mathematical sense.]

256 SCHEMES AND SYMBOLS. [Book II.

can be repeated, the divisibility or indivisibility of the rest,
in a word the possibility of generating the whole series
of numbers by the use of these constant relations and
operations, or, as we should say, the possibility of exhibiting
every number as a function of other numbers. Things
then, ought also to have the same inner structure, their
series ought also to be arranged according to the same
principles, so that the nature of the one might be exhibited
as a function of the nature of the other.

But it is also asserted by Aristotle along with others that
the Pythagorean school declared that numbers were things,
or at any rate that things were numbers. Even this is quite
intelligible to any one who is acquainted with the history of
philosophic ideas and the customary ways of expressing
them. To a certain extent indeed the Pythagoreans would
have been right in making .this assertion, and this justifies
us in supposing that they actually made it ; for as already
said what they intended was by no means merely to apply
numbers to the quantitative determinations of things whose
real nature is independent of these determinations, e.g. you
may have similar triangles of very various sizes : their
numbers were meant to signify that which distinguishes the
essential character of one thing from the essential character
of another ; a was a because its content was constructed
according to a the function-form or the generating law of
one symbolic number, and was thereby distinguished from b
which was b because it followed j3 the generating law of
another symbolic number. It was quite possible then to
say, with a reservation to be presently noticed, that the
essence of a thing, in the sense of that which distinguishes
it from another thing, lies in the number immanent in it

The other assertion that the essence of things, in the
sense of that in virtue of which they all are things, or their
reality, consists in these numbers, or that numbers are the
real things, was perhaps not positively made by the Pytha-
goreans in this form : if they did make it, they certainly

Chap. III.] NUMERICAL SCHEMATISM. 2*7

could not justify the latter expression, but they could as-
suredly justify the former : for if there is actually nothing
whose nature is not determined by one of these symbolic
numbers, the numbers are assuredly the conditio sine qua non
of every reality; to treat them as more than this, and to
speak of the numbers themselves as the real things, is an
unwarrantable straining of language, though we shall pre-
sently see how prone to it the thinkers of all ages have
been.

There remains one great imperfection which we have
already mentioned. The same typical series of numbers
has to repeat itself in a number of parallel series of actual
things, m a l> c d, /3y6, a b c & ; how then are the mem-
bers b ft b distinguished from one another if the whole
nature of each of them is exhausted by the same symbolic
number? To this there is no answer possible: at this
point the theory, which aimed at embracing the nature of
things completely, relapses again into a mere application
of a general law of structure to various cases whose charac-
teristic differences must be regarded as given. But this is
what makes it serviceable for our present purpose as an
illustration; it thereby becomes an attempt to frame a
universal scheme for the relations of kinship and
difference between all the groups formed by kinds of
content that can ever by any possibility come to be con-
sidered.

189. In order to justify the length of this discussion I
would point to the extraordinary tenacity with which this
desire to find a scheme for the whole contents of thought
has maintained itself through the course of ages. It showed
itself first in this form of mystical speculations about
numbers; over these we may pass very lightly; as such
speculators were satisfied with anything however meaningless
so long as it was interesting and startling, they were, to
speak plainly, always in search of a secret truth which they
never found, and it must always have needed a very sym-

LOGIC, VOL. I. S

258 SCHEMES AND SYMBOLS. [Book II.

pathetic hearer to find in the symbols a better expression
for the meaning put into them than could have been
obtained without them.

Presently the speculators ceased to found their dreams
on this purely arithmetical basis and wandered away in
various directions. In the first place every discovery made
by advancing science that has any important bearing upon
the relations of things has almost without exception been
extended into a scheme for the articulation of the whole
world. For a long time people traced everywhere the
behaviour of the four elements of the ancients ; and in
later days the mystic significance of this number four did
not pass away, it was only transferred to the newly dis-
covered constituents of organised bodies, 'carbon, hydrogen,
oxygen, and nitrogen; it agreed admirably with the four
quarters of heaven, for zenith and nadir of course fall
outside our natural line of sight ; it agreed equally well
with the four seasons of the temperate zones, within which
these speculations were carried on, and with the four in-
dispensable cases of nouns ; at a later date, as the theory
of astronomy came to completion, the contrast between
centrifugal and centripetal tendencies entered into men's
notions of all things and was fused into one with the
opposition of the sexes and the relation of acid to alkali ;
the discovery of magnetism and electricity caused the
scheme of polarity to be carried even further if possible
into the consideration of all conceivable things.

Other speculators proceeded in the opposite direction,
starting from the just reflexion that even the relations of
numbers are, in part at least, only instances of other still
more abstract fundamental relations ; these then (they hold)
must be sought, and will be found if we simply reflect upon
the operations by which our intellect does in fact arrive at
its ideas of all things whatever. Now every idea, or at least
every compound idea, is made by setting down an a, dis-
tinguishing from it or opposing to it a b, and finally bringing

Chap III.] GENERAL SCHEMATISM. 259

both into a relation c thus thesis, antithesis, and synthesis
come to be regarded as the scheme upon which all reality
is constructed and as the rhythm which thought must main-
tain in the orderly consideration of that reality. But it is
easy to see that the more abstractly these symbols are con-
ceived the more they pass over into notiones communes
which do indeed apply pretty well to everything but give
us no adequate knowledge about anything. Logic then
meets all this wild talk with the demand that things be
considered, divided, and investigated simply and solely
with reference to their several natures, for there is no
universal scheme that can be applied, and the employment
of merely fanciful models can only injure the impartial
quest of truth.

19O. Of this unfavourable verdict I can abate nothing,
and in some remarks which 1 wish still to add I have no
such intention. When the content M of a conception, an
idea, or a perception is given to us in such a manner as to
unite in the form ju, a number of characteristics, or parts, or
points of relation, it is a quite justifiable scientific curiosity
that prompts us to enquire how the examples of M will
behave, how they will be altered and distinguished from
one another, when we vary within the allowable limits either
the parts of M only, or both them and the general form of
union \i.

In the first place if we keep to the former kind of altera-
tion, there will usually be but little interest in tracing all
the kinds of M that are got by simply changing the quantity
of the characteristics, for these kinds will, in most cases at
least, resemble one other and only repeat the same thing
on a different scale. But if one of these characteristics m
be of such a nature that for it the opposition of negative
and positive has a plain and palpable meaning (such an
opposition for instance as there is between right and left,
attraction and repulsion, concave and convex, and generally
between ascent above a zero-point and descent below it)

S 2

260 SCHEMES AND SYMBOLS. [Book II.

t

then it concerns us greatly to know what happens to M
when we substitute m for + m in its generating law.
Supposing y ~ f x is the equation of a curve, we always
take the trouble to set down in turn the positive and
negative values of x, and not till we have united the results
thus obtained do we think we have arrived at the nature of
the curve, which in this case presents itself to our per-
ception not as a mere generality, but as the whole which is
got by combining every possible example of the general
equation. If we happen to see, in a piece of ornamenta-
tion, a volute which bends downwards to the right, our
imagination is stimulated in a similar way ; even if we have
no mathematical knowledge of the generating law of this
curve, we understand, by reason of the homogeneousness
of directions in space, that the volute might be repeated
in a precisely similar though opposite bend upwards to the
right, and again with another opposition upwards to the
left and downwards to the left. If now these continuations,
suggested by the beginning which we see, are not carried
out, though the surroundings do not give any obvious
reason for this incompleteness, our aesthetic feelings are
unsatisfied, but this demand for symmetry has also a
logical foundation. It is of the very essence of a law that
it shall apply to all variations of the points of relation which
it comprehends ; there is therefore a contradiction in a
perception which suggests a law together with the possibility
of its prevailing universally, and yet actually presents it as
prevailing only in part : what we miss in the perception
appears as a defect in the thing : we supply it in order to
remove the groundless want of universality.

We always feel a similar impulse in examining con-
ceptions. Whenever in any M one of its determinants
may vary from -f m to m, which it can only do by
passing through the intermediate value m = o, the tripartite
division thus suggested becomes for us a scheme^ which we
take as the basis of our investigation of the whole extent

Chap. III.] SCHEME MERELY SUGGESTIVE. 261

i

of M. This is the point which I wish here to emphasize,
in order to mark the difference between this proceeding and
the wild dreams we have just condemned, viz. that this
scheme can be nothing but an invitation to turn our enquiry
in a particular direction^ and cannot give us by anticipation
a picture of the result at which we shall arrive. It does not
always happen, as in the case of the volute, that the
counterparts we expect can be found : whether the change
from + m to m gives other possible kinds of M at all
depends upon the nature of the form of union /x. Still less
can we see beforehand whether the kinds thus obtained will
be in any way proportional to the differences of the con-
ditions, and if so in what way : it is quite possible that for
a certain JJL this absolute opposition of + M and m is
absolutely meaningless. Our method then will be to let ju,
likewise pass through all the possible forms given by the
various alternatives ; here also for mere additions of quantity
we shall expect only a series of similar results, but for every
cardinal point at which p, takes a qualitatively different
significance or passes at a bound into its opposite we shall
expect a quite new formation to appear in M which depends
upon /x ; and lastly for every remarkable feature which we
find in a special case of M we shall expect to find as
counterpart an equally remarkable feature in a similarly
conditioned special case of a similarly constructed N (as
for instance when we find that waves of light behave in a
certain way we look for corresponding behaviour in the
waves of sound) : but all this remains only a question put
to the object, to which we await the answer : the answer
which enquiry yields may turn out quite contrary to what
we expect, but must be accepted whatever it be. Where
those dreamers deceived themselves was in supposing that
whenever their scheme which they assumed to be universal
was applied to any matter whatsoever, every place in it
would always be filled by some remarkable form of that
matter, none would ever remain empty, and further in

262 SCHEMES AND SYMBOLS. [Book II.

supposing that as these various matters, passing through

the same sequence of changes, filled up the several places
of the scheme, the forms which filled the same places would
by a striking resemblance or analogy in their whole character
announce themselves as connected, as akin to or as coun-
terparts of one another. When this was not the case, there
was a strong temptation to try to fill up the gaps by ground-
less suppositions, and to restore the desired symmetry in
the corresponding members by giving undue prominence
to secondary features.

191. Among modern attempts to unfold in a scheme the
meaning of the world there have been some grand ones
which even seemed to avoid an essential fault of the Pytha-
gorean theory. In another work (' Geschichte der Aesthetik
in Deutschland/ p. 176 ff.) I have examined at length the
motives which led to the development of the Hegelian dia-
lectic, the most important of these attempts ; I will content
myself here with making a few remarks on its logical char-
acter. The Pythagoreans in conceiving development in
countless parallel series with different contents took no
count of the differences by which the corresponding mem-
bers of the various series are separated from one another in
spite of their occupying the same place in the general
scheme. The decimal system, with its ascending powers of
the number ten, never led them, as it might w^ll have
done, to treat these parallel series as themselves successive
periods of one and the same main series, resembling
one another in their internal structure, but raised one
above the other so to speak by the height of the level
at which they exhibit this structure, like the octaves in the
musical scale.

The imagination of the modern philosopher has supplied
this deficiency; the many parallel series are contracted into
a single series, composed of cycles of similar structure, the
last member of each cycle making a starting-point of a dis-
tinctively new character for the development of the next.

Chap. III.] HEGEL'S SCHEME OF DEVELOPMENT. 263

If it is possible to find the first member of the whole series
and the law which determines the form of the first cycle, the
variety of the contents which form the members of the fol-
lowing periods may be explained by their distance from the
starting-point and the transformation which the initial mem-
ber has undergone at each step of the way. Hegel then
requires us to concede as a metaphysical presupposition, of
whose correctness logic cannot judge, that the world is no
sum of things that stand and events that go on one beside
the other, the former standing quiet till they are stirred to
change by a stimulus from without, the latter determined in
their inter-action and in their whole course by universal
laws that hold good always, but that instead of this all the
variety of the world is only the development of a unity that
never rests, all events only stages in this development or
secondary effects of it, and things themselves but appear-
ances, either transitory or begotten anew at every moment,
whose whole being lies in the active movements of that
unity, crossing each other and coming to a focus in them as
subordinate vehicles of that development.

In this account of Hegel's point of view I make no pre-
tence to unimpeachable accuracy, which it would be difficult
to attain in a long exposition and quite impossible in a short
statement ; but what has been said is enough to enable us
to understand that within each dialectic cycle these different
forms, whose significance somehow constantly increases,
cannot simply occur one beside the other, but that each
must issue out of the preceding one : development, in short,
is the very essence of the system.

102. Now no development is imaginable without a defi-
nite direction which it takes in contrast to others which it
does not take ; but it is equally clear that in. this case above
all others it is impossible for the unity which develops itself
to receive this direction from without; it must be determined
by the nature of that unity itself. But here we find that no
accurate and exhaustive expression can be obtained for the

264 SCHEMES AND SYMBOLS. [Book II.

entire nature of that which under the name of the absolute
is regarded as the one basis of the world, but that what we
mean by it in a sort of presentiment is fully revealed to us,
nay comes to be completely itself only in and through the
development, indeed, the very name indicates this, for as
it is nothing but development, it cannot be itself before it
has begun to develop.

The only point of departure then that is left for us is this
fact itself, i.e. the knowledge that the absolute is not rest
but development. Assuredly then its development must
take that direction and form which follow from the concep-
tion of development itself, and which therefore must recur
in every example of the conception. This opens up a very
simple line of thought. If any A is to develop itself, it
cannot already be that into which it has yet to expand itself ;
neither can it not be, or be void of content, for then it
would not be the determining ground of that which is to
be ; as yet unexpandcd and shapeless it must still be the
determinate possibility of its future growth, in a word it
must be ' in itself ^ ' or potentially that which it is to become.
But its nature would not consist in development if it were
to abide in this potential state ; it must actually become
that which it is its nature to be able to become. But
becoming or the process of development is only an inter-
mediate step between possibility and fulfilment ; as merely
coming to be, hovering between starting-point and goal,
that which is developing itself would be neither identical
with itself as it was in its potentiality, nor yet already that
which it has to become. This at once enables us to see
why the second stage of the development, in which that
from which we started is as it were divided against itself,
was called by Hegel ' other being ' or ' being otherwise 2 ' ;
we see it still more clearly when we remember that it is to
the ground of the whole universe that this unfolding is in
strictness ascribed ; the process of its becoming does not
1 [' An sioh.'] 2 [ Anderssein.']

Chap, TIL] THE THREE STAGES, 265

consist in a simple movement in a straight line, but in the
generation of an infinite variety of forms, of which it was
the possibility; each of these is one of its results, none ex-
presses its whole nature ; the sum of all may indeed contain
a complete expression of this whole nature, but only for the
observer who adds up the sum and combines this manifold
into a unity in his thought. But that which is developing
itself must be this unity not only for others but for itself, if
it is actually to become that which it was its nature to be-
come ; and thus the name of t being for self l ' is given to
this third stage of the cycle, signifying the completion of
becoming, the attainment of the end of development, the
return of the potentiality into itself. This return of course
is not a simple return ; i.e. we do not mean that the inter-
mediate stage of the process is set aside 2 without leaving
any result behind or wiped clean out ; it must be set aside
in the sense of being stored up and preserved; the last
stage, being for self, is richer than the first, the potentiality,
by the history of the process through which it has come
into being.

It is easy to find images for this; thus the octave of
the initial note is a return of the latter into itself, and
yet preserves in its heightened pitch the result of the in-
tervals through which it has passed ; thus when a mind,
in which universal truths were innate in the form of
methods which its thought instinctively followed, had, by
passing through various experiences and enquiries, in-
volving doubt and the removal of doubt, arrived at a full
consciousness of these truths, it would merely have returned
to itself and yet would be enriched. I will forbear however
to explain in detail the peculiar meaning of these phrases ;
for us it is enough that in the third stage of the develop-
ment something is given which is indeed a consequence
of the first stage, yet is not identical with it but opposed
to it as actuality to possibility.

1 [< Fursichsein.'] 2 [' Aufgehoben.']

266 SCHEMES AND SYMBOLS. fBook II.

Thus understood the three moments or stages of ' being
in itself,' ' other-being,' and c being for itself,' are but the
component parts of the conception of development, and we
shall be able to recognise them in everything that develops
itself. But Hegel's system rests, as we said, on the con-
viction that the whole content of the universe, the whole
intelligible world, i. e. both nature and mind, are but stages
in the development of the one absolute, and that within
each of these great provinces the several members proceed
in the same rhythmic order, each founded upon and issuing
out of that which goes before, and that accordingly the sum
of all that is intelligible and all that is real would present
itself to us if we knew it completely as a great series, whose
several periods are similarly constructed but have each
a peculiar significance in its content which is ever rising
higher and higher. Upon this conviction we do not here
intend to pronounce any judgment ; but it remains for us
to ask what is the logical value of the dialectic method just
described.

193. It is easy to see that it is not strictly speaking
a method in the sense of a direction how to find something
that we are in search of; it is rather a scheme^ in the sense
in which we have used the word above, which only invites
us to enquire if anything is to be found in a given direction
or in a spot already marked out, and if so what it is, though
of course it implies a confident expectation that the search
can never be in vain. If we try to apply this scheme to the
independent treatment of a generic conception M, in order
to arrange its various species in a series corresponding
to their essential resemblances' and differences, or if we try
by means of it to exhibit in their true relations to one
another a series of conceptions which are connected by
a variety of other circumstances (as e. g. right, wrong, crime,
and punishment are connected), we at once find how un-
certain it leaves us as to the direction in which our thoughts
are to be turned. It is possible that this uncertainty might

Chap. III.] THE THREE STAGES OJ< RIGHT. 267

vanish if we could appeal to a complete philosophy which
had already set down in a universal series the history of
the development of all that is thinkable, and had therefore
arrived at a conception of right so perfect as to reveal
at once the direction of its further dialectical development.
But to say this would be to deny from the beginning the
applicability of the method as a universal direction for the
discovery of truth ; it can prove itself such only by this
independent service which we require ; i. e. it must be able
merely by means of its form of procedure to teach us how
to develop any given conception in all its proper con-
sequences.

Suppose then that we have given us the general concep-
tion of right, for evidently the other three that we named
refer to this as a primary conception already fixed : what
now is it 'in itself or potentially? into what 'other-
being ' does it pass over ? into what * being for self ' does it
return ? It is at any rate evident that a right involves
an estimate of relations which prevail between the claims of
various persons to exercise their wills upon some object
which brings them into collision. It follows that there can
be no right if there be no world with relations and objects
for the exercise of will, or if there be no persons who can
direct their wills to the same ends in one and the same
world. Right then is only potentially right and not yet
that which according to its conception it is to be, so long as
it only denotes by anticipation the approval or disapproval
of relations which do not yet exist.

Its * other-being ' is also quite intelligible ; it all comes to
the simple truth that general conceptions mean nothing
when there are no particulars for them to connect ; the
( other-being ' of right consists in the various rights whose
conditions lie in the existence of this nature, of these human
personalities with these definite wants and claims ; after the
general doctrine which sets forth the conception of right
will come the special doctrine which contains its applica-

268 SCHEMES AND SYMBOLS. [Book II,

tions. This direction is so simple that we do not need
to wait for the dialectic method to teach it to us ; but that
method does not help us in the least to carry it out ; for
after all experience alone can teach us what conditions
do in fact exist which give occasion for the development
of the general idea of right into special forms of right.

194. There is, however, yet another kind of advance that
we can conceive. ' Other-being ' certainly does often mean
the passing of the universal into its various particular
forms ; but I have already remarked that the Hegelian
doctrine lays stress upon the relation of opposition which
prevails between the two members, including the opposition
of the universal to the particular : this idea of opposition,
universalized and carried to its extreme pitch in the concep-
tion of contradiction, gives a further meaning to other-
being,' it may stand for the simple contrary of that which
the first (the being in itself) stands for. In pursuance
of this train of thought, right was made to pass into wrong ;
and wrong was made to issue in punishment, not indeed as
the ' being-for itself/ but as the means of reasserting the
violated right by the negation of its ' other-being/ i. e. of
the crime.

Now here again we have nothing that would not be just
as clear by itself without all this apparatus of the dialectic
method; and further, the method is actually confusing.
Any unprejudiced person would say to himself on reflexion
that all right has living reality only when living persons not
only know it but respect it in their actions, but that the
movements of men's wills are not in fact governed by
the ideal which they ought to follow ; wrong and crime
therefore appear, not as something necessary that must
exist, but as something possible that may y and indeed
always will^ exist, to judge by what experience teaches
us of human nature. In the transition which the dialectic
method gives there is none of this cautious bridging of
the gap between the two conceptions; it is represented

Chap. III.] HEGEL ON CONTRARIES. 269

as part of the very conception of right that it shall pass
over into wrong, and the paradox is not to be justified
by a plea which will be presently considered.

The transition to punishment as the third stage offends
us less merely because we supply the motives which are in
truth not given at all by the method itself. The method
does indeed demand restoration of the right, and that
by negating its negation the wrong; but it does not tell
us by what procedure this task, stated abstractly as the
negation of the wrong, is to be carried out. Why should it
take the shape of punishment? The evil disposition out
of which the wrong sprang is equally negated by disapproval
and by improvement, the harm done by payment of
damages, the violation of the dignity of the law by re-
pentance, and by a fresh recognition of its bindingness.
All these considerations show that the dialectic method was
of no use here except as a scheme, with places marked out
which we might seek to fill, but that, though we were
tolerably successful in filling them, the content with which
they were to be filled was only to be got from a quite
independent examination of the peculiar nature of the
object in question.

195. We said that it seemed to us absurd to maintain
that it is part of the very conception of right to pass over
into wrong ; but this swinging round of a conception into its
opposite has been so often and so emphatically claimed as a
higher truth discovered by dialectic, that it is worth while to
return to the point. Hegel remarks 1 that at first of course
the understanding fancies it can apprehend the nature and
truth of the real world by a number of fixed conceptions
complete in themselves and exclusive of each other; but
that the truth is that different conceptions do not simply
stand one beside the other with equal claims to represent
the finite, but that the finite of its own nature does away
with itself, and passes over of itself into its contrary. Thus
1 [Vol. VI. of his collected works, p. 152 f.]

270 SCHEMES AND SYMBOLS. [Book II.

c

we say that man is mortal, regarding death as something
whose ground lies merely in external circumstances ; and
according to this view man would have two distinct pro-
perties, that of living and that of being mortal also. But,
according to Hegel, the true way of regarding the matter is
that life as such contains the germ of death, and that in a
word the finite in itself contradicts, and thereby does away
with itself.

Here we can detect, more readily than we can in some of
the other passages in which Hegel treats of dialectic, a
confusion between two different statements. It is to the
conceptions by which we try to apprehend reality that fixity
and completeness are attributed in the first sentence : it is
not the conceptions but the finite thing to which we apply
them that is said to pass over into its contrary, and in
this latter statement lies all the truth that the passage
contains, which truth is shown by what follows to have been
uttered unintentionally or even contrary to the intention of
the author. For when the finite as such does away with
itself, it does so not because the general conceptions which
apply to it have lost their definiteness and swung round into
their contraries, but because it, the thing to which those
conceptions are applied, as finite or as actual, is unable
permanently to fulfil what is required of it by these concep-
tions, though each of them is true of it at one , moment ;
through a defect in its nature it passes out of the province
of one unchanged conception into the province of another
which is equally unchanged. But the conceptions them-
selves do not alter their eternal meaning because it is only
for one moment perhaps that they are a correct measure of
the changeable objects to which they are applied.

The true view of the matter then cannot be that life as
such bears in it the germ of death, and that the finite in
general contradicts itself: it is rather the two parts of this
statement that contradict each other. Life as such does
not die, and the general conception of life obliges the living

Chap. III.] A PHILOSOPHICAL CALCULUS. 271

thing to live, not to die ; it is only the finite, mentioned in
the second part of the statement, i.e. only particular living
bodies that carry in them the germ of death. And even
they do so not in virtue of the idea of life which is realised
in them, but assuredly only by force of external circum-
stances, i.e. only because that combination of material
elements through which alone life is manifested on the
surface of this earth is unable to exhibit an undying
example of life, though that would in no way contradict the
idea of life, whether this inability be regarded merely as
a result of the laws of nature which are here in operation, or
as part of a universal plan.

Similarly right never itself passes over into wrong, but
sometimes the will of a living person which ought to
embody it may, through want of judgment or through the
impulse of passion, be led into wrong while striving to do
right, and sometimes the law, which, men being what they
are, could not be administered at all if it allowed exceptions,
may do a wrong in a particular case involving complications
for which no provision has been made.

Logic then can in no way accept this doctrine that con-
ceptions dialectically do away with themselves : but the real
world as we find it is so arranged and ordered that what is,
though it does not do away with itself, yet docs of its own
nature pass from the province of one conception into that
of another ; and the fact that we find it so is worth notice,
as a fact about things that is to say, not as a peculiarity of
the intellectual tools by which we come to know them.

196. In any case, even apart from all the objections here
raised, the dialectic method would in the end give us only
an arrangement of our conceptions, an arrangement which
might no doubt present various points of interest to persons
fond of reflecting and comparing, in the aesthetic impression
produced by the discovery of analogies, parallels, and
contrasts, but which would scarcely open up a new way of
knowledge that could lead to definite new judgments or

272 SCHEMES AND SYMBOLS. [Book II.

propositions, or to a better and more precise settlement of
questions hitherto doubtful. To supply this want which the
dialectic method fails to supply is precisely the aim of other
vast attempts, viz. the attempts to found a logical language,
a universal mode of characterising conceptions, or a philo-
sophical calculus, at which Leibnitz laboured so long. The
mere addition of a series of large numbers would be an end-
less task if we were obliged to have a distinct image of each
one of the thousands or hundreds of units composing them,
and to build up each of these numbers separately and at
last their sum by repeatedly adding unit to unit. But our
system of ciphering enables us, without the need of
distinctly forming even any collective idea of the numbers,
to set units under units, tens under tens, hundreds under
hundreds, and then, by adding up each of these simple
columns, unerringly to bring out a result which itself in turn
we are quite unable to represent adequately in a single
picture by any effort of our imagination.

Now our conceptions so far resemble numbers that they
also contain for the most part a variety of individual images,
whose union with each other is not distinctly before us at
every moment, but only thought of in one collective im-
pression ; but they are denoted by words far less perfectly
than numbers are by figures. By the use of words that are
akin (though we are often no longer conscious of- the fact)
speech does indicate the kinship between contents, but very
imperfectly, for kindred ideas are also denoted by inde-
pendent roots : the kind of kinship between them is no
less imperfectly expressed, for the small variety of ways in
which derivatives may be formed is quite inadequate to the
manifold relations that have to be indicated ; moreover,
instances of each relation occur which, as the first to take
the fancy of the framers of language, are denoted by simple
words in which the characteristic derivative form is wanting ;
and finally the name of a conception never gives us all the
ideas that make up its content marked by simple signs and

Chap. III.] LIMITS OF SYMBOLIC EXPRESSION. 273

i

united in such a way that when we have to combine several
conceptions M N O we may shut our eyes to the meaning
of the whole and apply ourselves to combine some of the
component ideas with the same certainty of arriving at new
and correct results that our system of ciphering gives us in
numerical calculations.

These defects of language then we are called upon to try
to amend ; we are to dissect all our conceptions till we have
found the simple primitive ideas of various kinds which
admit no further analysis and the simplest ways in which
they can be combined, and we are to characterise these
by fixed signs, in order to obtain by their combination a
symbol for each conception which shall adequately express
its content. We need not think that the object of this
undertaking is the formation of a new speakable language,
which could never supplant the national and historical
forms of speech : its result would be a collection of for-
mulae for the purposes of scientific thinking only, to which
recourse might always be had for the settlement of the
doubts which arise from the employment of ambiguous ex-
pressions : for Leibnitz flatters himself that if we once got
such an instrument disputants would always cut their
quarrel short by an amicable agreement ' Let us reckon
it out.'

197. This is no doubt one of those enterprises whose
execution alone can finally decide whether they are prac-
ticable ; it would be over-hasty to deny the possibility of
that which might after all perhaps be realised by a happy
invention. However, the utter want of success hitherto
makes the inherent difficulties of the task more evident for
the present than the possibility of overcoming them. If all
we had to do were to make a system of signs for marking
the contents of our conceptions, the problem might appear
difficult but not insoluble. For then we should probably
begin by passing over all the generic conceptions of natural
history and limit ourselves to those conceptions whose

LOGIC, VOL. I. T

2 7 4 SCHEMES AND S YMBOLS. [Book II.

t

union in thought leads to difficulties which impede science
or the practical deliberations of life. Nevertheless even
this problem is harder than it seems, and the possibility of
solving it derives only an apparent confirmation from the
mathematician's language of signs and the symbols of
chemistry.

It is characteristic of the mathematician that he reckons
only with comparable elements, with magnitudes, the sim-
plest combinations of which he certainly can symbolise
quite clearly and unambiguously; but as the functions and
equations thus obtained grow more and more complex, we
see more and more plainly even here a sort of deterioration
in their employment. In the place of denominations which
really exhibit the inner structure of the magnitude in ques-
tion so as to indicate quite plainly how they are to be
treated in the calculation, we find introduced in order to
secure the necessary conciseness arbitrary symbols which
no longer have this property, but resemble the words of
ordinary language whose meaning must be known quite
independently of their sound. The expression V i still
expresses the origin of the function for which it stands, and
from this we can determine by general rules what results
when we multiply it once or several times by itself: but
this expression has already been discarded as too lengthy
and replaced by the other expression / which asc it stands
gives no clue to its signification, and whose meaning must
be otherwise already known if it is to be used correctly.
When we go on to speak of B-functions and r-functions,
these expressions are certainly concise, but we can only
understand them by representing them as equivalent to
other lengthy formulae, which in turn are only made in-
telligible by a previous explanation of the meaning to be
attached to the general signs of magnitude and symbols of
combination employed in them. All this is no reproach to
mathematics, nor is it any proof of the impossibility of a
universal system for characterising conceptions; it only

Chap, III] RULES FOR USE OF SYMBOLS. 275

shows that any formulae that the latter could give us would
not by themselves tell us all we need know, but would pre-
suppose a great deal which we should have to learn before
we could even understand them.

The symbols of chemistry make this still plainer : as yet
they refer only to the quantitative relations of the combining
elements, and to some extent to the supposed form of their
union what letters are to stand for the several elements,
and how their sequence is to denote the arrangement of
those elements, we must of course learn or know by heart,
as both can only be determined by convention : but no one
can tell merely by looking at the formula thus constructed
whether it stands for a gas or a fluid or a solid body, nor
what its density is or its specific gravity, nor what its colour
may be, whether it is fixed or volatile, soluble in water or
insoluble. If a man after looking at the formula answers
these questions correctly he does so upon the basis of
analogies with which his experience supplies him, and which
he could not draw from the formulae themselves with any
certainty that they would be correct. And yet all that is
wanted here would be the determination of properties or
modes of relation, which though not absolutely homo-
geneous are yet as physical processes dependent upon one
another and functions of one another, and therefore give
room for t hope that laws may be discovered which will make
it easy to mark by signs their dependence upon each other :
but the difficulties would be vastly increased when we tried
to characterise all our conceptions and had to deal with the
combination of unhomogeneous elements which yet have a
necessary relation to each other.

198. But it is not a system of signs only that we want,
nor is the success of mathematics due to its symbols,
though the skill with which they have been chosen has
no doubt furthered its advance. The truth is that the
usefulness of the signs rests here upon the fact that we
already have unambiguous rules, which enable us to deter-

T 2

276 SCHEMES AND SYMBOLS.

mine what follows from the simplest combinations of
magnitudes, and then being applied anew with the same
freedom from ambiguity to the results thus obtained issue
in these elegant and certain methods of solving problems.
It is these rules that we must feel the want of when we
try to combine conceptions which denote something more
than magnitudes so as to produce a certain result; and
I believe that we have absolutely no reason to flatter
ourselves with the hope that these rules would of them-
selves suddenly become perfectly clear so soon as we
had analysed into their ultimate constituents the essences,
contents, and matter to which they were to be applied.
Assuredly there is no need to insist on the fact that
increased clearness in the objects cannot but have a
favourable effect on the certainty of our conclusions re-
garding them; but in the main it is not by analysing
our conceptions and tracing them back to primary con-
ceptions, but by dissecting our judgments and tracing them
back to simple principles that we must hope gradually
to fix our convictions which on so many points are still
in flux.

But there are two things which we shall require to know :
first what are the necessary consequences which follow
from certain definite relations which, as we either arbi-
trarily assume or are forced to believe, hold between the
contents of various conceptions ; and secondly what general
laws, not proved to be necessary but found to hold good
in fact, connect various ideas in such a way that our
reason, founding upon these laws, can deduce the con-
sequences that will then necessarily follow from given
conditions. These problems, which concern the application
of the form of judgment, we must for the present attempt
to solve without the valuable assistance which that uni-
versal system of signs would no doubt afford if it were
once completed.

Note on the Logical Calculus.

THE idea of a logical calculus has been often taken
up and often abandoned : but the Englishman Boole has
recently made an elaborate and careful attempt to carry
it out, which is beginning to attract attention in Germany
as well as in his own country. Though I freely admit
that the author's ingenuity makes his able work 1 very
charming, I am unable to convince myself that this calculus
will help us to solve problems which defy the ordinary
methods of logic.

Boole does indeed insist that the result of a calculation
when completed must be expressible in logical terms ;
but he holds that between the statement of the problem
and its solution a course of operations may be introduced
whose several steps allow of no logical interpretation ; and
he appeals to the extension of mathematics by the intro-
duction of imaginary quantities. This appeal is hardly
relevant. The mathematician could not avoid imaginary
formulae : he lit upon them in the course of well-founded
calculations : he has always sought for the interpretation
of the enigmatic expression and has actually found it in
the province of geometry. In the logical calculus on the
contrary this working in the dark to which recourse is had
from time to time would have to take place by means
of symbols which have been arbitrarily chosen to denote
logical elements and the relations of these elements. If
therefore a calculation is really of use only when it allows
us to solve single problems mechanically, without requiring
us to be conscious at every moment of the logical meaning

1 [' An Investigation of the Laws of Thought/ London, 1854.]

278 NOTE ON THE LOGICAL CALCULUS. [Book II

of what had taken place, it becomes all the more necessary
that the rules which make such labour-saving processes
possible should be determined upon purely logical prin-
ciples without any rash and misty analogy from the province
of mathematics. Though on this point I entirely agree
with the admirable exposition of Schroder 1 , yet I cannot
entirely follow him : his demonstrations, which after the
manner employed by mathematicians follow upon the
statement of the theorems to be proved, have in my opinion
no significance beyond that of establishing that the whole
calculus is consistent with itself and that all the trans-
formations and combinations of its elements which it allows
lead to the same results when applied to the same pro-
blems : but we can only feel confident that the calculus
as a whole is applicable, when it has been directly shown
that each universal proposition is only the transcription
of a logical truth into the symbolic language that has been
adopted.

It has long been the custom in the section of logic that
deals with artificial classifications to make use of letters to
denote the marks which combine in various ways to form
the different species that fall under a concept. Supposing
that the three marks ABC belonged to the general notion
My the principle of disjunction would direct us to reduce
each of them to its subdivisions a^ # ? # 3 ..., h l b^ <f> a ... ; the
complete set of triplets of the form a b c, of course not
counting repetitions or permutations, would represent all
the kinds of J/, which, failing any closer determinations,
may be regarded as equally possible. These groups ob-
tained by combination express per se merely the simul-
taneous presence of their elements ; they leave the nature
of the connexion between the latter undetermined in two
respects.

P'irst of all they do not assign the final form which is to
be the result of the completed combination. Where logical
1 [' Der Operationskreis des Logikcaldals,' Leipzig, 1877.]

Chap. III.J NOTE ON THE LOGICAL CALCULUS. 279

classification is aimed at this want is supplied by the image
which is retained in thought of the abstract M, of which the
kinds are in question ; this M is to be added in imagination
to each combination a b c, as the general outline which the
union of the elements is to fill in ; apart from such an
occasion for the procedure by combinations, a b c taken by
itself only designates any object of thought, no matter how
constituted, in which the marks , b, and c are found to-
gether, or what is more important, any case, which it has
not yet been possible to characterise more closely, in which
the conditions a, />, and c are found together. This uncer-
tainty does not exist in mathematics, for the form which the
result of the calculation is finally to take, is here completely
and solely determined by the definitely assignable nature of
the connexion which this science requires to be introduced
between its elements.

Now with regard to this second point also, the reciprocal
determination of their component parts, the formulae em-
ployed in the combinations, in themselves, contain no
explanation of any kind. In algebra custom has made
them an expression of multiplication ; the particular sign of
this operation which has to be retained in the case of arith-
metic has been found unnecessary, in the case of algebraic
calculations at least, and the product of multinomials has
been found equal to the sum of the combinations of their
elements. Logic, on the other hand, does indeed pre-
suppose every mark that belongs to a whole to be connected
in a particular way with every other, but it has no means of
actually expressing these specific determinations, and en-
trusts them to our independent knowledge of the subject.
But what universal laws it does possess on its own account
with regard to the connexion of the marks bear no re-
semblance to the idea of multiplication. I will not here
lay much stress on the fact that the multiplier, which must
be thought of to begin with as a whole number, leaves the
value of the multiplicand as a separate number unaffected

28o NOTE ON THE LOGICAL CALCULUS. [Book II.

and only repeats it several times over ; while every mark <:,
which is annexed to a combination a b> not only modifies
the reciprocal determination of these original elements, but
at the same time by adding to the matter of the thought
limits the extent of its application. Anyone who cared to
dispute the question might perhaps find it easy even on this
point to make more of the analogies between the two sets of
relations than of their differences. But it is an essential
fact for our purpose that while multiplication is forced to
retain both the recurrences a a, b b> and the permutations
a b, b , as indispensable components of its product, logic
can admit no meaning in the former and no distinction
between the latter. Thus the nature of the case presented
no occasion for departing from the neutral significance of
combination-formulae which can have many kinds of mean-
ings, and applying to them the mechanism of calculation,
which has strictly speaking no suitability to them except as
symbols of quantities that can be multiplied. It could only
be ventured on in the hope that the more extended applica-
tion of the calculus would compensate, by results which no
other means could attain, for a cumbrousness inevitable at
the outset, seeing that exceptional rules were necessary to
bring such an inappropriate mode of calculation into har-
mony with its logical object-matter.

Every A, according to the law of Identity, must = A.
Natural thought has no motive to determine such an A
over again by a characteristic A, in the same way in which
A would be determined by a second mark b. No doubt we
speak of a human being as truly human, or emphatically of
a man who is indeed a man ; but we only employ such
expressions where it is permissible to distinguish the con-
ception M of an ideal from the conception /jt of the particular
facts from which the realisation of the ideal is expected.
At bottom, therefore, we are not determining a single M by
itself. The human being M /ut that is thus pronounced to
be truly human, corresponds to its determination M once

Chap. lll.i NOTE ON THE LOGICAL CALCULUS. 281

only and then completely, and just so in another aspect
corresponds to its zoological conception ^ once only and
then completely; such a thought bears no resemblance to
the attempt to determine quadrupeds over again by repeat-
ing the character ' quadruped.' Nothing but the machinery
of the calculus can suggest the requirement that a should be
determined by a as in multiplication ; but then the formula
a a = a or # 2 a which is now introduced to restore logical
truth, should at least abstain from professing to be a newly
discovered fundamental law of thought, or indeed anything
but a make-shift contrivance to correct an improper proce-
dure. The determination of a by a is logically speaking an
operation that cannot be performed ; it is only because and
in as far as, in the context of our thoughts, such a fruitless
attempt does not result in cancelling the a on which it is
made, that it is permissible to substitute a by itself for the
a' 2 to which the calculus would bring us ; but by no means
to treat this a 2 as existent, and pronounce it equal to a.
The left side of this equation contains an insoluble problem;
the right contains, not the solution, but what has to be ac-
quiesced in because there is no solution.

This is no mere verbal dispute, as may be seen from
some considerations which Boole subjoins. If we accept
a" = a for an equation, it is an easy step to the inferences
a' 2 a = o or a a* = o; Boole resolves this last formula
into a(ia) = o. Now the law of excluded middle
teaches us that everything that is thinkable is either a or
not a ; this truth is expressed by Boole, who indicates the
totality of the thinkable by the symbol i, by saying that
not-tf is what remains of this totality when we subtract a
from it; so that (i a] is the contradictory opposite of a.
Now the meaning of giving the equation the form in which
one side is zero can only be that the combination on its
left side has no extensi6n that falls under it, and cannot
therefore occur at all. Thus the formula a(ia) = o
becomes the expression of the law that nothing thinkable

282 NOTE ON THE LOGICAL CALCULUS. [Book II.

can be at once a and not-a. We may be delighted with the
plasticity of the calculus which furnishes such a graphic
expression of a familiar truth ; but we shall be the less
prepared to admit the interpretation which Boole gives his
formula on p. 50 of his work. It shows, he contends, that
the law which is regarded as the highest principle of meta-
physics is only a consequence of a law of thought which is
really mathematical in form ; that it is because this law
finds expression in a quadratic equation that our divisions
and classifications have to be performed by dichotomy ; and
that if the equation had been of the third order we should
have been forced to proceed by trichotomy.

I am sure that I shall not be guilty of trichotomy in the
sense of hair-splitting if I object to this extraordinary piece
of argument. Boole himself mentions that from a 2 = a we
can further deduce a' 6 = a, but he disposes of this cubic
equation with the remark that two of the factors which it
presupposes, (i -f- x\ are incapable of logical signifi-
cance ; and it was clearly the same reason that decided him
at an earlier stage to attach his inferences not to a' 2 ~a = o
but to a a 2 o. This procedure implies an idea which
is quite correct; among the numerous formulae which
can be mathematically derived from the supposed logical
principle a* = a none have any meaning but those which
express something that is of use in logic ; the validity of the
logical law does not depend on the shape of the formula ;
it is the value of the formula as a symbol that depends on
its agreement with the import of the law. But the quadratic
form itself and its interpretation are altogether a mere
caprice. I shall not insist on the point that according to
a 2 = a, a should have been at once substituted for # 2 ,
which would have brought us back quite intelligibly to
a a o', for even if we believed it possible to retain a 2 as
a real result of a practicable determination of a by a, and as
such to equate it with a, still there was no sort of logical
justification for resolving a a 2 into a (i a). In mathe-

Chap. III.] NOTE ON THE LOGICAL CALCULUS. 283

matics, where we are speaking of magnitudes, the trans-
formation is correct and in it i really means unity ; but in
logic the difference a # 2 does not present the least
motive for regarding it as the product of two factors. The
i, which is introduced in doing so, is not unity, which it
would have to be if the resolution were to be mathematically
correct, but is Boole's arbitrary though not inappropriate
symbol for the totality of the thinkable ; the truth that a and
i a taken together exhaust this totality must therefore
be established to begin with, in order to so much as make
the interpretation possible by help of which the formula is
intended to yield it.

These chimeras have not found their way to Germany ;
but I have mentioned them at length because of their con-
nexion with a general conception which does meet with
some assent among us. We do not overlook the differences
between arithmetical and logical computation ; but there is
an inclination to the idea of a more general mathematical
calculus 1 , for which this distinction of object-matter would
be indifferent. And it is true that every single act of
thought, apart from the logical import of its result, admits
of many uniform repetitions, and the result admits of many
connections and rearrangements ; further, the notions of
equality, inequality, and opposition have significance even
where tfyey do not relate to magnitudes ; though what
consequences they have in such cases must of course be
determined for each sphere according to its peculiar nature.
Still, when it has been determined, when, that is, it has
been decided under the jurisdiction of logic, what result
must be derived from the combined or separate occurrence
of several acts of thought and their particular results ; then
the recurrences and inter-connexions of all these elements
may be embraced under the same rules of union, severance,
and arrangement which hold good of all that is recurrent
and that has number. Only the laws which are specifically

1 [' Eines noch allgememeren mathematischen Algorithmic ' ]

284 NOTE ON THE LOGICAL CALCULUS. [Book II.

logical and, like the law of excluded middle, govern the
formation of the actual elements which are to enter into this
new connexion, must stand on their own feet ; and it is an
idea as incorrect as it is confused to expect that they can be
established by any mathematics however abstract which
should still merit that name in contradistinction to Logic.
On the contrary, all that such a science would have to teach
would be the development of the simplest logical truths,
which are uniformly true of the manifold and its combina-
tions, whether those of what has number and is homogeneous,
or those of what has mere relations and is heterogeneous.
Many things may be proved by mere verbal deductions ;
and so it may be held an important task to reckon up these
truths, in their abstract form apart from their applications ;
I think it rather tedious than indispensable.

As direct expressions of such extremely simple truth we
at once think of the axioms, the separate introduction of
which is hardly more than a matter of form. Obviously the
logical calculus must agree that a a, and that every a and
b which are equal to a third thing c are equal to each other ;
only the definition of equality demands a few words. Logic
uses a to indicate a general mark, a general class, or a
general case ; and is therefore able to accept the language
of the calculus, according to which a is the symbol of a
class, whose extent comprehends all individual things or
cases of whatever nature which share the character a.
These relations of extent are all that the calculus notices ;
it therefore sets down two class-symbols, a and />, for equal
when they present to thought classes composed of identically
the same individuals and are therefore only two names for
the same class. In such a case a and b may be different in
themselves, even if their extensions are fully coincident ;
thus equilateral and equiangular triangles, if nothing but
their extension is considered, are of course merely two
names for the same class ; still in logic we could not pro-
nounce the two conceptions equal as regards the contents

Chap. III.] NOTE ON THE LOGICAL CALCULUS. 285

which they directly declare as their own meaning. It
follows just as simply from those simplest truths that it is
always possible to comprehend two acts of thought and
their results in a sum a + b\ that a b is also possible in
logic if the necessary homogeneity is obtained by b being
included in a ; that the other combination a b, which col-
lects the two characters into one idea, represents a new
class-symbol with a defined extension ; and finally, that
where the problem put before us is only that of carrying
out some uniform mode of connexion, no difference can be
made by the order of the summanda or factors which we
combine to make a sum or product.

These easy analogies between mathematical and logical
reckoning are less deserving of mention than the differences
which are derived from the specific nature of logical thought.
I have already mentioned the equation a 2 = a and not less
paradoxical is the form in which the law a-\-a = a veils the
logical truth that each universal conception exists once
only, that therefore every logical assertion about what
comes under such a notion is completely exhausted when
it is once thoroughly admitted of the conception itself, and
that no new truth can be obtained by repeating the same
process on the same object. Just so the theorems a + al> = a
and a (a + b) = a remind us that every assertion which is
once granted to be universally true of a is also true of every
species of a that is still further determined by any mark ,
and that therefore the mention of ab beside a remains in-
effectual, in other words, the former is ' absorbed ' by the
latter. It is only the improper employment of the sign of
equation that gives these theorems their appearance of
peculiarity; all that they really say comes to this ; wherever
the mechanism of the calculus would naturally lead to the
forms # 2 , a + a, a + ab, these useless incidents of its method
are to be replaced for logical purposes by a simple a.

More important is the extended use which the calculus
makes of the law of excluded middle ; for the principle of

286 NOTE ON THE LOGICAL CALCULUS. [Book II.

r

Duality, which appears at this point as a new law of
thought, conceals nothing more than this familiar law. - If
we use a' to designate the contradictory opposite of a, and
i for the totality of the thinkable, then we have, really as
equations, the formulae a-\-a' = i, according to which all
possible matter of thought is exhausted by a and not-#, and
a a' = o which declares the impossibility of a union of a and
not-tf. No further proof is either possible or necessary,
whether for these laws or for the remaining one that the
negation of not-a brings us back simply to a and not to any
third thing ; they are logical truths which have no doubt
received in those formulae a very clear and convenient
expression.

The old Logic had its chapters about immediate infer-
ence, conversion, and contraposition of judgments, and
endeavoured by help of this same law to pursue the content
of an enunciated judgment into its relations to judgments
not yet uttered. Boole in a more comprehensive spirit sets
before himself the problem of developing the different
and mutually exclusive divisions of the thinkable that may
be formed by the affirmation and denial of the concepts,
class-symbols, or elements of whatever kind united in
a judgment. If x and y are the given elements, and x* and
y their contradictory opposites, then xy, xy', x'y and x' y f
are evidently the four classes into which all that is thinkable
must be divided ; that is, the constituent parts of the com-
plete division which Boole calls the expansion or develop-
ment of the given relation between x and y. It is some-
what inconvenient that following mathematical tradition
he designates that relation between x andj as a c Function '
of the two,/(#, j/); logically such an expression can mean
nothing, unless it is understood as the definition or predicate
of some M, for then all the constituents xy, xy\ etc.,
might be deduced from the given connexion between x and
y, and with them the coefficients which would indicate them
as possible or impossible within the extension of M. Boole

Chap, III.] NOTE ON THE LOGICAL CALCULUS. 287

however employs for the moment the independent function
f(x,y] in order to develop out of it in an equally general
way the law of the formation of those coefficients. His
original equation x 2 = x, as he can find for it only the
two arithmetical analogies o 2 = o and i 2 = i, induces him
to make the assumption that the logical and the mathe-
matical calculus would completely coincide if these two
values were the only ones which any magnitude could
assume ; and conversely, he takes all mathematical opera-
tions to be permissible in logic, on condition that the
class-symbols to which they are applied are treated as
magnitudes which admit of these two values only. So
taking ax-}- bx' as the given function /(jc), and/(i) and/(o)
as the two values which it assumes if we take x = i and
x = o (x' always assuming the opposite values), it is shown
that/(^) may be obtained by the combination of the two
values : f (x) =/(i) x H-/(o) x. The same consideration
leads, in the case in which the given function contains
the two elements x and j/, to the formula :
f(x,y) =/(i, i) *y +/(i, o)*/+/(o, i)*>+/(o, o) #y,
in which the two bracketed values refer in their order to
x and y respectively.

If any stress is to be laid on this scheme of the logical
development of a function, it would have been easy to
establish , it in a less bizarre fashion. It must after all
be borne in mind that the zero which denies every magni-
tude alike, so that for every ;;/, o . m = o invariably, and the
unit which every magnitude contains as a silent factor,
so that for every ;//, i . m = m invariably, are exceptional
and not merely homogeneous with all other magnitudes
even in arithmetic. Granted that they rank as magnitudes
when considered by themselves, still in combination or
multiplication with other magnitudes they have the general
logical import of affirmation and negation. What was
required in the above theorem was only this logical mean-

288 NOTE ON THE LOGICAL CALCULUS. [Book II.

ing, valid indeed for arithmetic but not derived from it;
it was therefore improper to give currency to the illusion
that logic is indebted to the peculiar laws of arithmetic
for the instruments with which it operates. I will take two
examples to show what I mean.

First, if M = a x + 1 x' , the value of the right side will
obviously be reproduced if we first suppress the first term
and leave the second, then suppress the second term and
leave the first, and finally add together the two that are
left:

a x 4- b x' a x 4- o . b x' -f b x' + o . a x ;

then the coefficients can of course be expressed by/(i) and
/(o), and

Again, let the function f(x, y}~ax-{-by be given and its
development with reference to the terms xy, xy' , x'y and
x f y r required ; and further, to make sure of what we are
speaking, let us regard/ (x,y) at the same time as a definite
M, whose definition, or specification of extent, is contained
in the right side of the equation.

Within this J/, the combination xy is possible in three
cases, being the a #'s which are also y, the by's which
are also ^, and the a x's which are also by in full, or
the ^ys which are also a x in full ; for none of these
combinations are expressly excluded by the right side of the
equation. We should therefore get a x y, bxy, abxy;
but as logically speaking the a b are included besides both
under a and under #, it is sufficient to exhibit a + b as
coefficients of xy\ apd of course a-f ^=/(i, i) is equal,
that is, to the value of the right side for x i,y = i. The
second term of the development would contain xy' } the
equation tells us that if we suppress by which can never
be combined withj/ there can occur within the compass of
M no y or not^v besides a x ; consequently a is the co-
efficient of xy f , and a of course =/(i, o). Just in the

Chap, in.] NOTE ON THE LOGICAL CALCULUS. 289

same way it follows that within M there can be no other
x f or not-# but by \ consequently b x f y is the third term,
and b of course =/(o, i). Finally the equation tells us that
the extent of M is entirely exhausted by a x and by^ and
contains nothing that is neither x nor y ; hence o is the
coefficient of x* y f , and it again =y (o, o).

Thus there is no doubt that the proposed formula of
function-development can be justified from purely logical
considerations, and I would attempt to establish this on more
general grounds if I saw more clearly what is the purpose
of the whole proceeding. The first examples which Boole
gives can only be regarded as exercises. If clean beasts x
are according to the Jewish law those which divide the
hoof y and chew the cud #, and then the development tells
us there are no clean beasts which divide the hoof but do
not chew the cud, and none which chew the cud but do
not divide the hoof ; that again there are no cleitn beasts
which do neither the one nor the other, and lastly there can
be no beasts which do both and yet arc not clean ; I have
my doubts of the frequency of the logical desire to go
through these deductions of given fact ; but if any one feels
the want, it is beyond a doubt more easily satisfied without
a calculus than with one. But there are two other problems
which Boole hopes to solve by help of such use of formulae ;
first, if a number of elements are given in any combination,
the equation which expresses this combination is to be
solved with reference to any of its elements at pleasure ;
and then it is to be possible to eliminate any one from the
equation, in order to display the relations of the rest to one
another.

As regards the first problem, I can only regret that Boole
abandons himself recklessly to his principle of permitting
himself all operations of reckoning if only their result can be
logically interpreted. From the given proposition 'All menj>
are mortal xj he obtains by contraposition * No man is not-
mortal' yx'=o. Now as x' + x = i arid therefore #'= i .r,

LOGIC, VOL. I. U

290 NOTE ON THE LOGICAL CALCULUS. [Book II,

we get y (ix) = o or yxy~o, xy~y } then further
#=- and by development of- we obtain x=y+- (i~j)

or =y + -X; this he takes to mean, introducing the mathe-

matical significance of the symbol - : ' mortal includes all

men and an indefinite number of what is not man.' Results
that could only be obtained in such unwarrantable ways
would certainly form no extension of Logic. Moreover, in
this case such arts were not even necessary. For not the
contraposited form 7(1 #) = o but the original y x
should have been employed, only with the precaution of
providing x from the beginning with a particularising factor
77, y = v x. The proposition 'All men are mortal' means
simply this and nothing more in the world besides ; it merely
regards y as subordinate to an x within the compass of
which there is something else as well. There is no possible
meaning in finding over again by calculation precisely what
was presupposed, and what is self-evident, that is, that x
comprehends beside the v x which are y a further in-
definite number of kinds which are not y \ that therefore

With respect to the process of elimination, I shall con-
tent myself with giving an example. Every logical equation
can, by applying contraposition to the affirmative judgment
which it expresses, be reduced to the form in which one
side is zero ; for the equation x z = o simply means that no
x is z. I pass over Boole's doctrine about the procedure of
collecting ail given single judgments or equations into one
solitary resultant equation, and suppress the scruples which I
feel as to the necessity or productiveness of such an operation.

It is granted then that the equation is to be presented in
the following arrangement; pab + qab' + ra! b+sa!b r = o ;
then the product of the coefficients pqrs equated to zero,

Chap. III.] NOTE ON THE LOGICAL CALCULUS. 291

is assigned as the result of the simultaneous elimination of
# and b. This is easily seen with the ordinary appliances
of Logic. For logically this equation cannot have the value
o unless each of its terms taken by itself o. Further,
pab o says that No/ a is b ; but qab f o gives by contra-
position All g a are b, and so in Cesare, No q a is / a, or,
_pga o, and this again gives No/f is 0, or by contraposi-
tion, All / q are a'. Again r a' b = o gives, No r a' is b ;
but .$ a' b'~ o gives by contraposition All ^ a' are b ; so we
get in Cesare, No s a' is ra', or, s ra f o, or, No r j is a'.
If we subsume the second conclusion No r s is a under the
first All p q are a\ there follows in the same figure, No r s
is / q ov pgrso. It is easy to see that if a similarly
arranged equation with one side zero contains besides a, b,
and a ', b 1 ', more such pairs of opposites r, /, the elimination
may be continued in the same way. But no doubt for such
cases there is value in the abbreviated rule that the result of
the elimination consists in the equation of the product of
the coefficients to zero. If the equation had contained
besides a term z = o independent of the pairs to be elimi-
nated, it would persist without change, and might be added
to the preceding term, so that in the result p q rs + z = o
each of the terms by itself is = o. Schroder remarks on
this question at p. 23 of his work that the results of the
elimination of a symbol a from several isolated equations are
less comprehensive than those of an elimination from the
combined final equation; xa+ya'=o and / a + q>a' = o
when taken apart, only give xy o and/ ^ = o; while on
the other hand the combined equation gives xy+$x+
py +p q = o ; and for this reason he thinks the latter order
of procedure preferable. Is he not in this artificially
making little difficulties, simply out of the order of procedure,
which must ultimately depend on the development of the
functions? Why are we forced to unite the four terms
x a = o, y a? = o, p a = o, and q a' = o, although they must
be true by themselves, in two equations, instead of regard-

U 2

292 NOTE ON THE LOGICAL CALCULUS. [Book II.

r

ing them as four terms to be employed at pleasure ? Then
we might find without difficulty all results of elimination
which we had any interest in ascertaining.

I do not maintain that the same syllogistic process will
easily bring us to our goal in every case, especially in more
complicated cases. But Boole himself insists that we must
carefully analyse what we mean in every case, before trans-
lating our notions into the language of the symbols ; and
I certainly believe that the fulfilment of this postulate
would enable us to dispense altogether with the calculus,
and that Logic would prove rich enough to allow of the
invention of adequate means of solution corresponding to
particular problems, even if these means were not stereo-
typed beforehand. With reference to this point I mention
a problem which Boole 1 puts and which Schroder repeats.

It is assumed to be known from an analysis of experience
that in a certain class of natural or artificial products the
combinations of the marks a b c d e are subject to the
following rules ; and in such a way, that not only the
occurrence but also the non-occurrence of each particular
mark belongs to the conditions from which the presence or
absence of the others has to be inferred.

1. Wherever a and c are absent at the same time, e is
present, together with either b or d, but not with both ;

2. Where a and d occur, but not e, b and c will either
both be found or both be missing.

3. Wherever a is found in conjunction with either b or e
or with both at once, either c or d will be found, but not
both together.

4. Conversely, where, of the pair c and d, the one occurs
without the other, a will be found in conjunction with either
e or b or with both at once.

It is required to ascertain :

i. What can be inferred from the presence of a with
reference to b, c, and d ;

1 [' Investigation of Laws of Thought/ p. 146 ff.]

Chap. III.] NOTE ON THE LOGICAL CALCULUS. 293

t

2. Whether any relations, and if any, what, exist between
, t, and d, independently of the other marks ;

3. What follows from the presence of b with respect to
tf, <:, and d^ and

4. What follows for a, c y and d independently of the
other marks.

Boole anticipates that no logician would find the right
answers to these questions by syllogistic process, unless he
knew them beforehand ; I fully admit this, but who would
be tempted to select that process for attacking this problem
while the more suitable one offers itself spontaneously?
We" have only to make a list (it is a purely mechanical
process) of all the combinations of five which can be formed
out of a I c d e and a' // c f d' e', avoiding repetitions and
the inclusion of contradictory elements, and then, or while
making the list, to suppress those which are excluded by
the totality of the given conditions. This leaves only TI
combinations ;

abcd'e ab' c d' e a' b c d e a'b'cde
abcd f <f ab'c'de a' b c d e' #' b' c d e f
abc'de ab'c'd'e' a f b c f d' e

From these we can read off the answers to the questions
proposed :

i J . We infer from the presence of a that either c or d
is present, but not both, or else that <*, r, and d are all
wanting.

2. There is no independent relation between <, c^ and d,
for all conceivable combinations of them with b', c', d' are
equally realised.

3. From the presence of b it follows that either a, t,
and d are all absent, or some one alone of them is
absent.

4. If a and c are both present or both absent, d is im-
possible.

Similar questions about e which are not proposed could
1 [Cp. Boole, pp. 148-9.]

294 NOTE ON THE LOGICAL CALCULUS. [Book II.

be answered out of the same conspectus without any
distinct operation.

I borrow from Schroder's treatise for purposes of com-
parison no more than the beginning of the solution by
calculation; not so much to show that if all the inter-
mediate terms are actually supplied it is by no means
distinguished by brevity, but chiefly with the general object
of elucidating the use of the calculus by help of an instance
that involves a real problem, and is not merely going back
upon what we know to clothe it in awkward formulae.

By contraposition of the positive judgments which con-
stitute the given conditions of the possible combinations,
and so reducing them, as equations, to the form in which
one side is zero, we obtain

from i. a! c f [>'+ bd+Vd*] = o;

from 2. a d \b c f + b' c\ e'= o ;

from 3. a \b + e] [cd + c'd'] + [ad'+Sd] [a'+fS] = o.

As the questions ask nothing about e and e' the first
operation to perform is the elimination, which we dispensed
with, of this pair of opposites. According to the rule given
above its result consists of equating with o the sum obtained
by adding those components of the equations which are free
from e and / to the product of the coefficients of e and /.
Now to begin with, the coefficient of e in 3. = a (c d + c' d'\
and that of e' in i. 2. and 3. = a' c' '+ a d\b / -J- b f c] -f
V [c d f + c' d} -, the product of the two is according to the
above-mentioned rules = a b' c d, and with the addition of
the terms free from e and /, which are = a f c' [b d + b' d']
f a b \c d + c' d'} + a' [cd'-\- c' d] the entire result of the
elimination would have to be brought together into

a \cd + b c'd] -f a' [c d'+ c ' d + b' c' d'} = o.
Now to answer by this result in the first place the second
question, about the relations between , c, and d, we should
have to eliminate a and #'; but the requisite product of
their coefficient is = o because each individual product as

Chap, in.] NOTE ON THE LOGICAL CALCULUS. 295

it arises takes independently the value o owing to the

combination of contradictory elements ; the result is there-
fore = 0, and we must accept this as a sign that there is
no independent relation between these three marks. How-
ever, we see at once that if we give the symbol p to the
coefficient of a that of a f will become/' or not^ ; we there-
fore obtain from a p -f a f p' == o the two equations ap = o,
or No a is /, and a' p f o, or No not-0 is not/. The
first of these gives at once ; all a are not/, or p' \ hence
a = c d' + c' d + tf c f d', which formula answers the first
question.

I omit the continuation which would be needed to answer
the third and fourth questions, and confine myself to re-
marking that in the whole of this problem no use has been
made of the development of functions, of the importance
of which I expressed my doubts above ; the required
equations were obtained directly from the given proposi-
tions, and the eliminations out of them were conducted on
a method, the origin of which we explained to ourselves by
help of syllogisms in the second figure. Thus there is
nothing to be said against the appropriateness of the present
method ; but just as little against the superior simplicity
and plainness of that which we adopted. This, by the way,
had not to wait to be discovered by Jevons, for it was
already fprthcoming in the doctrine of classification, which
long since required in the first place the tabulation of all
the marks in their combinations, and then the cancelling
of all combinations that become inadmissible on taking
account of the reciprocal determinations of the marks. I
cannot therefore convince myself of the advantages to be
derived from the attempt to systematise in a fixed logical
calculus all the means of vivid and abbreviated presentation
to which every one has spontaneous recourse in given cases,
applying them with variations adapted to the proposed
problem. It is inevitable that a symbolic method intended
to make uniform provision for every case should purchase

296 NOTE ON THE LOGICAL CALCULUS. [Book II.

its suitability for the solution of one problem at the cost of
a useless prolixity in its treatment of others and of manifold
discords with the custom of language.

Even the quantification of the predicate, which was the
starting-point of recent English logic, was no new discovery,
but the superfluous inflation of a familiar idea to an ex-
cessive importance. That the predicate of a judgment,
except in case of simply convertible judgments, has a larger
extent than the subject which in part takes its place within
this extent; that therefore it is not merely the predicate
that determines the subject, but also the subject that
restricts the predicate to such a modification as is true of
the subject's self; these were old doctrines of logic, and in
its rules of conversion it went so far as to provide for their
application. It is true that the scheme of judgments gave
no special expression to this truth, just as the ordinary
linguistic form of the sentence did not. But what harm
was there in that, when the fact was known ? Did the want
of such an expression ever deceive a considerate thinker ?
And was it worth while, for the sake of amending such
trifles, to have recourse to such dangerous contrivances, as
to connect the natural expression of thought with a new
symbolism and a new calculus ? There could be no real
gain in expressing the proposition 'All men are mortal'
by y = v x unless a means could be discovered of defining
this v ; as long as it remains an undefined coefficient, it is
an ineffectual indication of what we knew before. In the
converse of this judgment c Some mortal is man,' the old
logic would bring to light this indefinite Particularity 1
neither better nor worse than that v would; if we object
to the expression 'some,' our objection might be easily
removed by the consideration that these indefinite particular
judgments are at the same time forms of modality, and
express the possibility of a conjunction of their predicate
with the general notion which forms their subject, by
1 [' Diese unbestimmte ParticuUiitat.']

Chap. III.] NOTE ON THE LOGICAL CALCULUS. 297

affirming such a connexion for some but not for all cases
of the notion.

There is a passage of Jevons (' Principles of Science,'
London, 1877, p. 59) which among others has occasioned
these remarks. He forms two premisses; sodium 1 = sodium
metal, and sodium = sodium capable of floating on water.
He draws the conclusion sodium metal = sodium capable
of floating on water. To this he subjoins these remarks.
" This is really a syllogism of the mood Darapti in the third
figure, except that we obtain a conclusion of a more exact
character than the old syllogism gives. From the premisses
* Sodium is a metal ' and ' Sodium floats on water ' Aristotle
would have inferred that 'Some metals float on water.'
But if enquiry were made what the ' some metals ' are, the
answer would certainly be 'Sodium 2 .' Hence Aristotle's
conclusion simply leaves out some of the information
afforded in the premisses; it even leaves us open to in-
terpret the c some metals' in a wider sense than we are
warranted in doing. From these distinct defects of the
old syllogism the process of substitution is free and the new
process only incurs the possible objection of being tediously
minute and accurate." Oh no ! we might admit the ' te-
diously,' but otherwise Aristotle is in the right. Jevons'
whole procedure is simply a repetition or at the outside an
addition^ of his two premisses ; thus it merely adheres to
the given facts, and such a process has never been taken
for a Syllogism^ which always means a movement of thought
that uses what is given for the purpose of advancing beyond
it. So the combination of words which he proposes is not
a syllogism at all, and consequently not one in Darapti.
The meaning of the syllogism, as Aristotle framed it, would
in this case be that the occurrence of a floating metal
Sodium proves that the property of being so light is not
incompatible with the character of metal in general. If he

1 [See Professor Lotze's Preface to the Logic.]
3 I* Metal which is Sodium/ Jevons.]

298 NOTE ON THE LOGICAL CALCULUS.

expressed this by saying c Some metal is capable of floating/

he intended of course not to repeat the premisses which
were known before ; but to enunciate the possibility of a
general distribution of this property among metals, as a
supposition whose correctness in fact there is ground for
testing further, since it is logically not inconceivable. Even
the expression ' Some metal ' is at bottom quite correct, for
Sodium certainly is some metal; the expression does not
enjoin us to think of other metals at the same time with it ;
it is true that it does not prohibit our doing so, but this
need not give rise to any error.

How often have modern enterprises like these proclaimed
the dawn of a wholly new epoch in logic, and the fall of the
contemptible system of antiquity! I am convinced that if
the ancient logic were to be really forgotten for some
generations and then rediscovered by some fortunate
thinker, it would be welcomed as a late discovery, after
long search, of the natural march of thought, in the light of
which we should find intelligible both the singularities and
the real though limited relevancy of the forms of logical
calculus with which we had made shift so far.

CHAPTER IV.

The forms of Proof.

199. IT was our business in writing of systematic logic
to enumerate the various forms of judgments and to point
out the precise mode of union which in each of these forms
is conceived as subsisting between S and P or as to be
effected between them : it is the business of applied logic
to consider what contents S and P can properly be joined
in one of these forms of union. Various problems which
we shall not always hold apart fall within this scope. In
the first place the communication of the thoughts of others
gives us numerous propositions of the form S is P y whose
meaning and purport is perfectly plain, but whose validity
is questionable : then there arises for us the problem of a
proof "far fat given proposition T. In the second place our
own observations may lead us to suppose that between two
ideas S and P there subsists a relation which if it were
known could be expressed in a judgment of the form S is
P : then we are called upon to discover the yet unknown
proposition T which would be the precise expression for
this supposed relation.

These two, discovery and proof, differ only in their
different use of the same materials. The same combina-
tions of thought by which the truth or probability of a
proposition J'was first discovered may always be applied,
when put spmewhat differently, and sometimes even with-

300 THE FORMS OF PROOF. [Book II.

out any such transformation, to prove the truth or prob-
ability of a given proposition T. Moreover it is easy to
see that the reflexion of the discoverer, if it is not to miss
its aim, needs at every step slight connecting links, re-
sembling a proof in form : and conversely that a proof will
never reach its goal without some inventive play of thought.
On the whole however discovery reaches farther than proof :
and so I will separate the two problems, though I shall not
always avoid the natural mixture of the two. Scientific
investigations lead to both in about equal measure; the
needs of life more frequently lead to discovery.

I find reason however again to divide the first part of the
subject, and to separate the proof of universal propositions
from the proof of particular or singular propositions. It is
true that a universal relation can seldom be established
between and P without the employment of knowledge
supplied by experience ; but as such knowledge, if it is to
lead to universal conclusions, must itself have universal
validity, we may regard it as knowledge which, though
originally derived from our experiences, is yet, now that we
have full confidence in its universality, to be counted
among the proper instruments of thought. The proof of
particular facts on the other hand, of historical events or of
the ordinary transactions of life, can never follow from
universal propositions alone, not even from such universal
propositions as are themselves derived from experience : it
presupposes the knowledge of a number of particular
circumstances, occurring only here and only here united in
this precise manner. The preliminary process of getting at
all these conditions, from which the conclusion is to be
drawn, requires peculiar instruments which we shall con-
sider presently. The solution of a proposed problem on
the other hand, even when the result is to be not a universal
proposition, but the establishment of a single fact, may be
connected with the proof of universal propositions : under
the conditions which here do not need to be sought but

Chap. IV.] SELF-EVIDENCE. 301

are given, and so far as they are given, the definite proposi-
tion T which satisfies them all is always to be found by
employing instruments of thought which are of universal
application, and these theoretical results are inaccurate and
need correction in practice only so far as we have failed to
state #//the conditions which 7* had to satisfy.

20 O. Every proof is a syllogism, or a chain of syllogisms,
which completes the premises required for the given propo-
sition 7] so that they fit into one another in such a way that T
follows as their necessary consequence. But the validity of
every conclusion depends upon the validity of its premises :
these again might be established by fresh proofs, but this
procedure would go on ad infinitum without any result were
there not a number of universal propositions which we
accept as immediate truths, which therefore neither need
nor are capable of proof, but are themselves the ultimate
grounds by appeal to which we may decide in every case
whether a conclusion is correctly or incorrectly drawn from
its premises. I do not intend as yet to discuss the question
of the source from which we obtain these immediate truths :
we are here concerned only with the mark which justifies
us in classing a proposition T among the axioms^ assent to
which we believe ourselves entitled to demand from every
sane person. Now it is conceivable that, just because there
is no possible proof of an axiom, this mark may in the last
resort be nothing but the self-evidence , the immediate
clearness and certainty with which the content of a uni-
versal proposition thrusts itself upon us as a necessity of
thought ; and in fact this is what we always come back to
in the end.

Experience however abundantly shows that propositions
which later generations have proved to be false, were as
self-evident to earlier generations and produced in them as
strong a conviction as any propositions whatsoever : rela-
tions which in the limited sphere to which our observation
is confined are seen to be permanently present or constantly

302 THE FORMS OF PROOF. [Book II.

recurring, without any contrary experience to disturb us,
very commonly assume the appearance of necessities of
thought. There is only one way of distinguishing the
spurious self-evidence of a prejudice from the genuine self-
evidence of a true axiom : we must try whether the contra-
dictory of T the proposition in question is as impossible in
thought as T itself seems to be necessary. This test will
often be quite decisive ; we shall often find to our astonish-
ment that the attempt to join S and P in the opposite way
to that asserted by the given proposition T leads to no
inner contradiction in our thought at all. In that case ^is
no axiom, but either altogether an error, or a truth that
holds true in some cases only, or a truth which though
universally true requires to be proved. In the other case,
when the contradictory proposition non-jT appears as im-
possible in thought as T appears necessary, we may with
greater confidence regard T as an immediate axiom ; but
the test does not even now give perfect security, for it is
quite possible that the inconceivability of non-7' and the
apparent necessity of T may both alike rest upon a spurious
self-evidence. Should these two simultaneous errors be
made, logic furnishes no short way of detecting them : our
mistake could only be gradually amended by our becoming
aware of the contradictions which experience offers to the
assumed validity of T, and by a slow and far-reaching
modification of our system of thought suggested by those
contradictions.

Such a double error will seldom be found in the case of
purely theoretical principles, more often in the case of the
principles upon which our moral judgments are based, and
which may be classed as genuine or spurious axioms,
although strictly speaking they do not seem to be necessities
of thought but only unquestionable truths, and their oppo-
sites do not seem to be unthinkable but only absurd.
That you ought to hurt your enemies was for a long time
generally accepted by the ancients as an unquestionable

Chap. IV.] THESES NOT WORTH PROVING. 303

maxim, and the opposite of it regarded as absurd : such
errors can generally be removed only by a gradual alteration
in men's habitual feelings.

201. Supposing now that T is a universal proposition
whose validity is not axiomatic, i.e. that it is such as to need
proof, we yet shall not set about proving it till we know
that T is worth proving. In three cases it will not be worth
proving. In the first place it will not be worth proving
if its content is an incomplete, and therefore an indefinite
thought. A man of untrained intellect, so long as he
confines himself to the objects which naturally come within
his scope, is usually conscientious in enumerating and
examining all the points which are important for the under-
standing of a fact : he follows the old rule which tells us to
ask 'quis? quid? ubi ? quibus auxiliis? cur? quomodo?
quando?' and to omit none of all these questions. But he is
quite helpless when he wanders off into general considera-
tions which belong to the province of speculation : he then
usually does not get beyond a clumsy expression for
something which he perhaps rightly believes, demands, or
assumes, but is unable to connect with any determinate or
determinable points. The philosopher on the other hand,
revelling in his abstractions, docs not always come to meet
him half-way, but often contents himself with employing
conceptions which when severed from their proper applica-
tion become utterly meaningless : the result is that vague
theses are nowhere so common as in the attempts of a man
who has had no logical training to philosophise by the light
of nature. That God and the world are one is a proposition
that no one can prove except him who propounds it ; so far
as his proof is correct at all it is the proof itself that tells us
what he meant by the proposition : any other person than
he who propounded it will, if he be wise, attempt neither to
prove it nor to refute it ; for that God and the world are in
some sense two is asserted by the proposition itself, for
otherwise it could not have distinguished them; but that

304 THE FORMS OF PROOF. [Book II.

they are also one in some one of the many senses of unity,
may be supposed without more ado.

That things are appearances is an equally ambiguous
proposition : the things which appear to our senses are so
of course, for otherwise they could not appear to us : but
that the things which though themselves inaccessible to
observation we suppose to underlie our sensuous perception
are also appearances is an incomplete thought till we deter-
mine what is to appear and to whom it is to appear. All
these and other similar propositions are not worth proof or
refutation, but are simply to be returned as they are to him
who brought them, just as in a court of law we refuse to listen
to a man who complains that he has suffered wrong without
saying what has been done to him and who has done it.

202. The second case is when though a perfectly clear
nominal definition may be given of S the subject, or P the
predicate of the proposition T 9 the definition contains a
combination of ideas which can be shown to be impossible,
or cannot be shown to be real. No one would trouble
himself to prove or to refute a proposition the subject of
which is a wooden iron : no one would seriously enquire
whether this wooden iron will burn in the fire like wood or
melt in it like iron. There is no such logical contradiction
in the ideas of ghosts and will-o'-the-wisps, but we defer
asking whether the former need sleep, and whether the
latter are attracted by buried metal, till their existence is
proved. What we here require may be called in general
the justification of a conception, which must without fail be
added to its nominal definition when use is to be made of it.

This may be effected in various ways. If M stands for
something which is supposed to have external existence, the
shortest way to justify M is to point at once to an instance
of it or to a fact in which its existence is given and
accessible to observation. If M denotes a combination
of ideas the validity of which means that it can be carried
out and that its result can be imagined or realised in a

Chap. IV.] CONSTRUCTION AND DEDUCTION. 305

mental picture, this very realisation of the content which M
demands, or in other words its construction will justify M
itself: thus geometry establishes the admissibility of the con-
ceptions it has formed by presenting in a visible form what
they till then only contained as a problem, thereby proving
most conclusively that the problem was soluble. If we can
neither point out any instance of M nor carry out its con-
struction, we must at least show cause or give a ' deduction'
which explains how in connexion with some demonstrable
reality or in pursuit of some problem we have been properly
and justly led to form this conception. Such a ' deduction '
cannot always directly prove the validity of M in the shape
in which the conception is presented, but it may always
show that M is a preliminary designation for some content
which we are reasonably and rightly looking for ; it remains
for the further enquiry whose beginning is hereby justified
to determine whether M itself can be justified as a valid
conception, or else how its content must be modified in
order to make it valid.

The ancients regarded the doubling of the cube as a
serious problem : but though they could not geometrically
construct the required line, whose cube should be double
of a given cube, yet it was all along certain that the problem
was soluble and that the required line was a magnitude
which coyld in some way be found. For it could be shown
that as the side of a cube increases its volume must also
continuously increase without any alteration in its shape :
among this infinite series of larger and larger cubes then
must be found that particular one which is double of a given
cube, and this implies that its side actually occurs in the
series of existing lines. We here show cause for the
necessary validity of that which is sought instead of actually
realising it in a construction.

Again it may be doubted whether one and the same con-
ception of length fits both curved and straight lines; but
setting this doubt aside it was not unreasonable as things

LOGIC, VOL. I. X

306 THE FORMS OF PROOF. [Book II.

then were to hope to find by a simple geometrical con-
struction^the straight line which is equal to the circumference
of a circle of given radius ; for it was certain that the length
in question depends upon the length of this radius and upon
nothing else. This hope was only banished by the com-
pletion of the enquiry, which showed that the circumference
cannot be expressed as a determinate real and algebraical
function of the radius. In the natural sciences a hypothesis
often assumes facts which we can never hope to establish by
direct observation : often indeed we must leave it to God
and the future to show even the possibility and constructi-
bility of that which we are for the present absolutely obliged
to assume. The only way of justifying ourselves in such a
case is to show from the given facts the pressing need of
the idea which we employ, with the reservation of course
that we may at a future time so alter it as to enable us to
construct it without impairing its usefulness. We shall
return to this point on another occasion ; for the present it
is enough to refer to the instances above employed as
showing what kind of justification is needed for conceptions
if their union in a proposition is to deserve proof or
refutation.

203. We will now suppose that the conceptions which
are joined in the universal proposition J'have the requisite
definiteness and validity: but even so we do not start in
search of a proof that shall exhibit J'as the necessary con-
sequences of premises that must be discovered, until we
have got some preliminary warrant that the proposition is
true as a matter of fact ; for it would be lost labour to
try to prove what is not even true. If T is a universal
proposition of whose field it is not easy to take a com-
prehensive survey, we first try whether T holds good in
some examples that lie near at hand : a single case in which
it did not hold good would do away with the universal
validity of T, and the problem would then be changed
into finding the conditions under which T has at least a

Chap. IV.] DIRECT PROGRESSIVE PROOF. 307

partial validity; if on the other hand that which T asserts is
found to hold good in all the cases of its application which
we compare, this trial, though being necessarily incomplete
it cannot prove that T is universally valid, may yet corro-
borate what it alleges so strongly that it will be worth while
to search for a proof. This very needful preliminary pro-
cedure, which will further on take its place among the
forms of proof, is in fact neglected but seldom, and that
mostly in cases where the validity of T 7 cannot be tested by
mere reflexion upon instances supplied by the memory,
but only by observation or experiment. The courtiers of
Louis XIII exhausted themselves in ingenious proofs of the
proposition that a living fish thrown into a bowl full of
water makes it overflow while a dead one does not, until
the gardener who was called in made the experiment and
showed the assertion to be entirely false ; but others make
the same mistake, and in the less exact departments of
natural science we frequently find subtle demonstrations
and explanations of phenomena whose actual occurrence is
entirely problematical.

204. Supposing now that this preliminary question is
settled, and that T is recognised as a universal proposition
that deserves proof, its truth or falsehood may be established
either in a direct or in a roundabout way, and this makes
the first division of proofs.

A proof is direct when it shows immediately that the
given proposition T is necessary or impossible ; it is indirect
(or apagogic) when it establishes the truth or the falsity of T
mediately by showing the falsity or the truth of its con-
tradictory non-7! In each case there are two directions
which the train of thought may take. We may call a proof
straightforward or progressive when it starts with that which
in the nature of the thing conditions something else and
makes that which is conditioned issue from it as its conse-
quence ; it is a backward or a retrogressive proof when it
starts from that which in the nature of the thing is con-

x 2

308 THE FORMS OF PROOF. [Book II.

t

ditioned in order to arrive at knowledge of that which
conditions it. The first form of proof, since it proceeds
a prindpio ad prindpiatum^ may equally well be called
deductive, though the opposite name inductive will not
be found so generally suitable for proofs of the second
form which proceed a prindpiato ad prindpium. And
finally there is yet another distinction applicable to both
these lines of proof : you may go forward (progressively)
either from general truths to T or from T to its proper
consequences, and similarly you may go backward (retro-
gressively) either from T's consequences to 7", or from T
itself to the truths upon which it is founded. We cannot
pronounce upon the comparative value of the eight different
forms thus obtained till we can consider each in reference
to the problems for which it is usually employed. The
following survey may enable us to do this.

205. The first form of proof, which is direct and pro-
gressive, proceeds from a universal truth, which is placed as
major premise at the head of the whole procedure ; in the
minor premise (or in a series of epi-syllogisms, if the proof
can only be completed in a chain of reasoning) it is then
shown in what relation S and P which are joined in the
given proposition T stand to that major premise; and
lastly the conclusion infers that by reason of these relations
of S and P the proposition T which was to be proved must
hold good. If the problem be stated in this general way it
seems as if all the three figures of Aristotle might be em-
ployed in this form of proof : the fact is however that the
first figure alone answers to the spirit of it. I do not reject
the third figure on the ground that as usually described it
only gives particular conclusions, while we here wish to
prove universal propositions ; if we put the particular con-
clusion 'some are P 9 into a modal form, 'that which is
*$* may be P,' we get a universal proposition which it may be
worth while to prove. For instance if we want to produce
an effect P, and have nothing to get it out of except an

Chap. IV.] ALTERNATIVE PROOFS. 309

unpromising material S, we shall be glad to see it shown by
a syllogism in Bramantip that S and P are compatible with
one another in the case of a subject M, and that therefore
S does not always make the desired effect P impossible.
But the third figure does not exhibit this proof in the
progressive form. It only states in the premises an instance
of the coexistence of S and JP 9 from which we may argue
regressively, ab esse ad posse, to their compatibility, The
second figure admits universal conclusions indeed, but only
negative ones ; these too may be valuable, but they cannot
be obtained by this figure without premises of opposite
quality, and therefore fail to satisfy us. For a universal
negative proposition 7J which simply denies a predicate P
of a subject S because S and P stand in opposite relations
to a third M, appeals to a mark which shows that S and P
cannot be combined, but not to a reason which explains why
they cannot : it merely expresses a fact which is indeed true,
but remains unintelligible till we have learned in an affirma-
tive proposition what S really is, and thus now can see that
because it is this it cannot be the other, viz. P. And so
the second figure, since it establishes its conclusions by
proofs which, though appropriate and convincing, give no
explanation, is also rather regressive than progressive in
character. And therefore under the head of direct pro-
gressive .proofs attention has usually been directed to the
first figure, especially to its affirmative moods, and for the
present purpose to Barbara exclusively : it is only here that
we find the subordination of a given idea under a general
truth, which enables us to understand not only that T holds
.good, but why it holds good.

200. This opinion is as old as Aristotle : it is worth
while to observe however that this form of proof is to be
regarded as an ideal in another sense than this : it cannot
fairly claim the praise bestowed upon it unless we succeed
in filling it with the content which its articulation requires,
i.e. unless we set down for major premise a general proposi-

310 THE FORMS OF PROOF. [Book II.

tion under which the special case of the minor premise
demands to be placed in virtue of its very nature, and which
theiefore would actually be the reason upon which the
validity of the proposition to be proved depends not merely
for our reflexion but in the nature of things. But it is clear
that we may use the form of this proof without in the least
satisfying this last condition. Many instances occur, and
that precisely in the field of mathematics where exact treat-
ment is required, of propositions that admit of various
equally convincing proofs all couched in this form of sub-
sumption, none of which therefore can claim exclusively to
express the proper connexion and development of the thing
itself. The possibility of presenting the same idea in very
various forms without altering its value enables us here to
subsume it under a great variety of universal major premises,
and to proceed from any one of these arbitrarily chosen
starting-points to the same assertion T. I am anxious not
to be misunderstood here and will therefore go into detail.

I will in the first place allow that we often find in mathe-
matics a proposition T which is so evidently only an appli-
cation of a definite major premise M that its deduction
from this major alone seems natural, from any other
artificial. I will remark in the second place that when T
may be deduced with equal ease from a variety of majors
M NO y I do not find in this alone any reason for saying
that these various proofs are foreign to the natural sequence ;
for it may be (though I do not propound this as the true
theory but only suggest it as a possible view) that the whole
of our knowledge (e.g. of geometry) rests in fact upon a
number of original and equally self-evident perceptions,
none of which can be deduced from any other, but which,
like the several components of one complete thought, are
each and all valid at once and connected in definite ways
with one another. We could then understand how in
virtue of this connexion the same proposition admits of a
variety of equally convincing proofs, according as we start

Chap. IV.] EX PL AN A TION AND CER TAINT Y. 3 1 1

from one or the other of those inseparably united percep-
tions : no one of these proofs will exclusively exhibit the
nature of the thing, but yet each may actually exhibit it in
the form in which it is seen from that particular point of
view ; the possibility of a variety of proofs rests in this case
upon the organisation of the content itself, which makes a
harmoniously articulated whole not on one line only but on
several lines at once. But I must nevertheless add in the
third place that there remain many propositions 7] whose
proof (I mean in this form of subsumption) can only be
effected by devices, which can be justified after they have
been applied, but to the application of which we cannot
find any invitation in the thing in question. It is to these
proofs, of which many occur in pure mathematics, and a far
greater number in applied mathematics, that the remark
above made is intended to apply ; though these proofs are
as conclusive as can be wished, it is yet quite beyond our
power to take them all in at one view, especially when they
form chains of many links ; and as they scarcely allow us to
do more than see the necessary consequence of coupling
each link to the one which follows, while the inventive in-
genuity which forges the chain seems to be guided by pure
caprice, we cannot honestly say that they show why the
conclusion T is true; they only constrain us to admit that
it is true. I have introduced this point because of its
practical importance. Our ideal of knowledge and demon-
stration no doubt is that we should deduce each given
proposition T from the determining grounds by which it is
in fact determined in such a way as to explain //, and not
simply assure ourselves of its certainty by a logical device;
and if this problem is to be solved, it can only be by a
direct progressive proof of this form. But it is soluble only
within narrow limits, and where it is not soluble, where
therefore we must content ourselves with the mere certainty
of T) this form of proof by subsumption has not the least
advantage over other forms. It is mere pedantry on the"

312 THE FORMS OF PROOF. [Book 11.

part of the logician to wish in spite of this to enforce it and
when a proposition can be conclusively proved in two words
by an indirect method to look about for a direct deduction,
which can only be effected by a chain of arbitrarily selected
links, which makes it a longer business to get to that
certainty, and which does not in the least help us to see the
reason why it is so.

207. A second directly progressive method of proof is to
start from the given proposition T, assuming it to be valid,
and proceed to develop its necessary consequences. If
among these consequences we find even one which contra-
dicts either established facts or recognised general truths, T
does not hold good as a universal proposition, and the proof
becomes a mode of refuting a given proposition ; it then
includes, as may easily be seen, that preliminary procedure
above mentioned, by which we assure ourselves before
entering upon the actual proof that among the given cases
there is no contradictory instance against the validity of the
proposition to be proved. If the development of the
consequences of T however far it be carried discloses
nothing inconsistent with known facts or truths, we have
not even yet got enough to establish the truth of 7J for the
next step in that development beyond the point at which
we have stopped, might reveal the existence of a contra-
diction hitherto concealed, but at any rate this procedure
suffices in the field of science to recommend a hypothesis,
which is then reserved for further examination. But the
true province of this method lies in practical life : it is the
method we employ to recommend proposals, arrangements
that are to be adjusted, resolutions that are to be adopted.
And here the incompleteness of the development of the
consequences is no obstacle ; in all human affairs it is
enough to ascertain what effects will follow from the appli-
cation of a proposed measure within such a limited time
and in such a limited field as we can readily survey : he
-who wishes to take count of all the subsidiary effects which

Chap. IV.] DIRECT REGRESSIVE. REPUTATION. 313

a microscopical examination might disclose, all the conse-
quences centuries hence of what we do to-day, is a super-
cilious pedant ; fresh measures will be taken to avoid
minor disadvantages, and the remote future must take care
of itself.

208. A third form, the first directly regressive form of
proof, proceeds from the assumed validity of 7"and works
back to the conditions under which this validity is possible.
The difference between this form and that just discussed is
not considerable, but there is a difference : it is not con-
siderable because the conditions requisite for the validity of
T can only be found by taking T as their basis and de-
ducing them as consequences from it, a procedure which
coincides with our previous direct progressive method : but
we see that there is a difference when we consider the
nature of that which is thus deduced. We may take as an
instance of both forms at once the ordinary way of solving
a problem in mathematics ; for every such solution is at the
same time a proof of the solubility of the problem, i.e. of
the validity of the combination of ideas contained by the
proposed problem T. If we assume that T is true and
develop the consequences which flow from it, these conse-
quences may be of various kinds ; some of them will be
particular circumstances which agree or disagree with given
facts, others will be general relations between various
objects which are either consistent or inconsistent with
truths otherwise established. If we only come upon
particular consequences which disagree with given facts
or secondary conditions, we may with certainty infer that
T does not hold, though we do not see the reason why it
docs not ; if T is a practical proposal, it may be that it is
quite acceptable in itself and that it is only its execution
that encounters some obstacle; and then we should have a
case of the second form of proof: if on the other hand we
come upon absurd general propositions which must be true
if 7'is to be true, then besides the certainty that T'is im-

314 THE FORMS OF PROOF. [ Book II.

possible we get also a strong hint as to the reason why it is
impossible ; that reason must lie in the general truths which
conflict with the absurd conditions we deduced, and herein
we find what this third form of proof does for us. It npt
only clears the ground for the subsequent discovery of a
direct and progressive proof of the contrary proposition,
but gives us a remarkably conclusive and palpable negation
of a given proposition T in the disclosure of all the absurd
assumptions that would be necessary if it were true ; and
on this account this regressive proof is often preferable to a
progressive one.

It cannot establish anything but the falsity of T, and so
remains a form of refutation. If in working backwards
from T we come upon none but admissible conditions, we
cannot infer that T is true except in mathematics ; for only
in mathematics is it possible to develop from a proposed
problem all the conditions necessary to its solution ; in
other cases we can never be certain that we have really
deduced from T everything without exception that is im-
plied as a condition necessary to its truth ; the next step we
took might bring to light an absurdity that we should have
to assume. Affirmatively then this method is in matters of
theory only able to establish the probability of T\ in prac-
tice however we use it to recommend a proposal just as
much as the foregoing progressive method. For when we
want to secure the acceptance of a proposal we not only
point out the consequences to be expected, but also show
that the conditions of its execution are not incompatible
either with the general requirements of justice and morality,
or with the means which are actually at our command. A
political measure always needs to be justified in these two
ways, after the former method by its useful consequences,
after this method by the admissibility, in the view of justice
and morality, of all that it implies : and in our daily life we
must take count not only of the advantage to be expected
from a provision, but also of the price wfe must pay for it.

Chap. IV.] DIRECT REGRESSIVE. COLLECTIVE. 315

209. A fourth method, the second direct regressive
method, starts from given propositions and proceeds to
prove from them the validity of T as the condition of
which they are the result. This is a line of thought which
we are very constantly impelled to follow : for the greater
part of our knowledge of general laws is won in this way by
reasoning back from given facts to that which must be
assumed as the condition of their possibility. It is easy to
see however that its most important applications belong to
the method of discovery which tries to elicit from that which
is given a T 7 which is as yet unknown. When the general
proposition T is given and we are looking about for the
several propositions which may serve to confirm it, the
proper method is always to begin with the progressive de-
velopment of that which as consequence of 7"must be true
if T be true : only when we have made a comprehensive
survey of these consequences do we proceed to compare
the result obtained with experience or with other truths, in
order to reason regressively from the truth of this result to
the truth of 7!

I will therefore postpone the consideration of much that
might be introduced here, and will only mention one species
of this method, viz. that which infers the universal truth of
^from its truth in particular instances, complete induction
or the collective proof. We are often compelled to employ
it : it is not always possible to prove at one stroke that a
proposition T holds good for all quantities, integral and
fractional, positive and negative, rational and irrational, real
and imaginary magnitudes ; but each of these several kinds
of quantities may offer some special point of attachment for
a proof that T is true of it ; if then we are sure that we have
included all possible cases of T, that is in this case if we
are sure that there is no conceivable kind of quantity besides
those named, then we know that T is true of all quantities
whatsoever. The general conception of quantity will then
no doubt contain some reason for this universal validity;

316 THE FORMS OF PROOF. [Book II.

nevertheless we cannot always point out this reason, or at
least we cannot always make it quite clear and self-evident ;
and then we must have recourse to the collective proof.

210. The necessity of including without any omission all
the kinds of cases to which T can apply if T is to be proved
universally true leads here to an interesting special form of
this proof. Mere completeness of course can always be
secured by dividing all the cases into say Q and non-<2, the
non-<2 again into R and non-7?, and so on as far as we like,
stopping say at U and non-7: but this is seldom of any
use ; for even if we easily find separate proofs for the posi-
tive kinds of cases QR 7, it is very difficult to find one for
the negative remainder non-6 7 " which embraces a miscella-
neous crowd of different cases. We are constrained there-
fore to take a case Q, for which we happen to be already
able to prove that T is true, and try to derive the other
cases R U . ., etc. from Q in such a way that it may be
evident that the changes by which Q passes into J?, and R
into U, either do not affect the conditions which made T
true in the case of Q y or else constantly reproduce them.
This is the method, familiar to mathematicians, first formu-
lated by Jacob Bernoulli^ of proceeding from n to n -f i,
chiefly applicable when the several cases in all of which T
is to be true form of themselves a series in which each suc-
cessive (n 4- i) th member is formed in the same precisely
definable way out of the preceding n^ member. If then it
follows from the way in which the member n + i is formed
from the member n that jTwhen true of the latter must be
true of the former also, it follows for the same reason that
it must be true of the member n -h 2, and so on for every
member of the series. For instance in teaching the elements
of algebra this method is usually employed to prove the
binomial theorem for integral exponents in a palpable way
by repeatedly multiplying the binomial into itself.

The general idea of this proof however is by no means
confined to mathematics, but is very often applied in com-

Chap. IV.] INDIRECT PROGRESSIVE. EXCLUSION. 317

mon life, sometimes under the not quite appropriate name
of a proof by analogy. In support of a plan or a statement
we first mention an instance in which the plan is evidently
advantageous, the statement obviously true \ then we show
that the other conceivable cases are in reality distinguished
from this case by no feature that could possibly make a
change in this respect ; and thence we conclude that T
holds good universally. It is easy to see how a careless or
sophistical use of this method may lead to error. Between
two very different cases A and Z we insert a great number
of intermediate cases, each separated from the next by an
inconsiderable difference d. Then instead of showing that
if Z'is true of A it must also be true of A+d> which is B,
we assume that it is so because d is so trifling ; we reason
similarly from B to C, and finally transfer the validity of T
from A for which it really held good to a Z which by the
accumulation of the many disregarded differences d has
become entirely unlike A and does not in the least belong
to the field to which T 7 actually applies.

211. The indirect methods of proof may be treated more
briefly. They bear formally the same relation to non- 7*
that the direct methods bear to T, and the only circum-
stance that makes them in some degree peculiar is that we
wish to arrive by them not at non- 7 1 but at T: they are
therefore not affirmative but negative proofs in respect of
non- T. The fifth method of proof, the first indirect pro-
gressive method, would have to show that non- T is false on
general grounds, and this may be done by syllogisms in the
first and second figures with a universal negative premise.
But we shall seldom find an opportunity of applying this
form of proof : if there be a direct proof for T we shall
prefer it ; if there be none, a universal refutation of non- T
is usually no easier.

The only form of this method therefore which is prac-
tically important is the secondary form, which in the place
of non-7 7 , the contradictory of T, substitutes the complete

318 THE FORMS OF PROOF. [Book II.

sum of all its contraries. As these contraries are all quite
definite positive statements, there is more hope of being
able to disprove each upon general grounds, and therefore
by a progressive method. The proof that non-7"is univers-
ally false which is formed by the union of these several
negative proofs is then evidently a regressive argument
corresponding to the positive collective proof. When T
and all the contraries of 7 T are conceived as together form-
ing the sum of all possible relations which can subsist
between S and P, the subject and predicate of T, the form
of proof of which we are speaking becomes that which is
known under the name si proof by exclusion: the truth of
.7' then follows from the falsity of all the other members of
this complete disjunction. One of the most important
applications of this form is the special case of a tripartite
disjunction, in which T has two contraries, i.e. in which
non-7 7 divides into two contradictories : then we get the
proof by the method of limits. We are familiar with this
proof and its very great importance in mathematics, where
it belongs equally to inventive and demonstrative reason-
ing : every magnitude a is either equal to or greater or less
than another magnitude d with which it may be compared :
if it can be shown that it is neither greater nor less than */,
the proposition a = d is proved. In practice this train of
reason generally takes another line: for the above statement
presupposes that our attention has already been directed to
the definite magnitude d which is proved in the end to be
equal to a. As a rule this is not the case, but we only
know that a is less than a second magnitude b and greater
than a third c : if then we can succeed in showing that the
same relation constantly holds as we diminish the value of
b to j3 and raise the value of c to y, the value of a must lie
between two limits fl and y which are constantly approaching
each other, and it will be possible to calculate this value
with an approximation to the truth which may be carried as
far as we please. The best known and most elementary

Chap. IV.] INDIRECT PROGRESSIVE. AD ABSURDUM. 319

example is the determination of the length of the circum-
ference of a circle by enclosing it between a larger cir-
cumscribed and a smaller inscribed polygon, and diminishing
the former and increasing the latter without limit by continu-
ally adding to the number of their sides. Such forms of
proof deserve our attention ; they are the potent instru-
ments by which we actually enlarge our knowledge ; the
development and application of this method by Archimedes
is a greater advance in applied logic than any that ever pro-
ceeded from the merely syllogistic art of Aristotle.

212. A sixth method, the second indirect progressive
method, would begin by assuming non-7} and proceed
to develop its necessary consequences, and then from
their falsity infer the falsity of non-7] the last step of
course being regressive. I will here refer the reader back
to the second direct progressive proof, and only add with
reference to this indirect method that it does not matter
how many true propositions may be deduced from non- T:
for it is quite possible for a number of true inferences
to flow even from a false proposition with respect to points
whose mutual relations are not affected by the error : but
a single false proposition which follows as a necessary
consequence from non- T does away with its universal
validity. If this consequence merely conflicts with given
facts tl^ere is properly no reason for calling this proof a
deductio ad absurdum, though the name is sometimes given
to all applications of this method : all that has been done
is to prove that an idea which in itself is not unthinkable
nor absurd is as a matter of fact untrue. But again absurd
or nonsensical is strictly speaking not that which is known
to be impossible in thought, but that which conflicts with
all probable suppositions, with our general feeling as to
what is true, and a number of truths involved in that
feeling, provable perhaps but not yet actually proved.
That 2 = 3 is more than absurd, it is impossible; but that
the whole world is a thoughtless jest, that parents should

320 THE FORMS OF PROOF. [Book II.

obey their children, that we should reward criminals and
be tender to sin, are absurd assertions. I would therefore
apply the name deductio ad absurdum only to the indirect
progressive proof which develops from non-7 7 consequences
which are not impossible in thought, but which are in-
consistent with a host of convictions accepted as truths
and sufficiently established. This kind of proof is very
constantly employed in daily life, especially whenever
non-7 7 states a thought, which is perhaps in itself correct,
in too general language, i.e. when it proceeds from too
wide a definition of the subject S to which a predicate
P is to be attached, or from too wide a definition of
this P. It is in this way that we prove the unreason-
ableness and foolishness of a proposed law, whether it
gives or takes away rights and duties, by showing what
further intolerable and monstrous consequences would
follow if the proposal were carried out universally. Usually
however the deductio ad absurdum is made to include
also that form of indirect proof which deduces impossible
consequences from an assumed proposition and thereby
refutes it.

A particular case of this is when the development leads
to a consequence which at once does away with the pro-
position from which we started, so that the inner con-
tradiction which lay in the assumption of its truth of itself
forces us to infer that it is false. As a simple instance
we may take the indirect proof of the proposition T that
on a straight line a b in the same plane and at the same
point c only one perpendicular c d can be made to fall.
Non-7 7 then would assert that several perpendiculars were
possible at the point c under the same conditions. Now
assuming that this is correct, assuming further that c d is
the first perpendicular, i.e. that it makes with a b two
adjacent equal angles a, any second perpendicular c e must,
in order to be distinguished from c d, make with it at the
point c some angle 6, while at the same time in order to

Chap. IV.] INDIRECT REGRESSIVE. 32 1

be perpendicular to a b it must make with it equal adjacent
angles. A look at the figure then is enough to show that
the two angles #-fdand a 6 must be equal, and each
equal to a right angle : but if a 4- 6 be a right angle, cr,
being a part of this right angle, is not itself a right angle,
which contradicts the original supposition that a is a right
angle. The equation -f- d= a~ d can only hold good when
6 = 0, i. e. when c e and c d coincide. The proposition T
therefore holds good : at the same point in a straight line
there can be only one perpendicular in the same plane.

We are constantly led to proofs of this kind when we
have to do with the simplest fundamental perceptions or
propositions concerning a coherent field of thought : the
impossibility of apprehending the relation of S to P other-
wise than as it is expressed in T, i.e. the fruitlessness
of the attempt to affirm non-TJ will always betray itself
by the fact that the consequences which follow from it
destroy or alter the subject S or the predicate P, which
were both assumed to be valid for non-7 7 in the same
sense in which they were valid for T.

213. The indirect proof, like the- direct, admits of two
regressive forms : these two, the seventh and the eighth in
our survey, have but little to distinguish them ; they bear
just the same relation to the falsity of non-7 7 that the
two direct regressive proofs bear to the truth of 7 7 .

The former (the seventh) method would work back from
non-7 7 to the conditions necessary to its truth, and then
reason back again from the falsity or inconceivability of
these principles to that of non-7 7 . In its application this
method differs but little from the corresponding progressive
method; for the principles which are necessary to the
truth of non-7 7 can only be found by taking non-7 7 as their
basis and developing them from it as its consequences, i.e.
progressively. The latter (the eighth) method would start
from given facts or principles and proceed to show that
they cannot be founded upon non-7 7 as their basis, but

LOGIC, VOL. I, Y

322 THE FORMS OF PROOF. [Book n.

rather expressly require the falsity of non-7! This also
we shall find can only be carried out by either developing
non-7 7 progressively into its consequences, and ascertaining
that if they held good they would make the given facts
impossible, or by taking these given facts for basis and
deducing from them, progressively as before, their necessary
presuppositions : but this will very seldom be of much
use, for in that case it will usually be easier to establish
directly that T as such a presupposition must be true,
than indirectly to establish that non-7 7 cannot be true,

I will conclude this survey with the general remark that
I believe that I have correctly distinguished in my classi-
fication the various aims of demonstrative reasoning, but
that not every one of these aims has corresponding to it
an equally important and equally peculiar form of proof,
clearly distinguishable from all the other forms; it was
enough therefore to examine in detail only those which
have in practice shown themselves to be methods that
are frequently applicable.

214. The reader will be surprised at the absence from
my list of the proof by analogy, I do not believe in its
existence. In all cases where we believe we can prove
by analogy, the analogy in fact is distinctly not the ground
of the conclusiveness of the proof; it is only the inventive
play of thought by which we arrive at the discovery of a
sufficient ground of proof : it is upon this ground, by means
always of a subsumption of the individual under a uni-
versal, that we establish the necessity of the proposition
to be proved. Although it will take a considerable space,
I think I must consider this point in detail.

It may be regarded as a fundamental principle of analogy
in the strict sense, holding good in all cases without
exception, that of like 1 things under like 1 conditions like 1
assertions are true, a statement which the mathematician
further expresses in a number of special ways adapted
1 [The German word is ' gleich' not ' ahnlich/' See note p. 327 below.]

Chap, IV.] ANALOGY. 323

to his various problems. It is easy to reduce this principle
to the principle of subsumption : if P is true of S under
a condition X, S and X may be comprehended in a general
conception M, of which as such P is true ; under the same
M we may subsume any other S which is like the first S
and subject to a like condition X; therefore the same
predicate belongs to this S as to the first. This trans-
formation, which may here seem arbitrary and superfluous,
cannot be dispensed with in the case of the second prin-
ciple, of unlike things under like conditions unlike as-
sertions are true. We may be inclined to regard this
also as unconditionally true, but difficulties thicken upon
us when we try to apply it. Suppose that unequal mag-
nitudes a and b are divided by the same third magnitude
c ; in this case the principle will hold good ; the quotients
will be unequal. But take a second case : divide each
of the unequal magnitudes by itself, and the principle seems
to fail; the quotient in both cases is i. Of course it
will at once be urged that the condition X y to which the
unequal elements a and b are subjected, is just not alike
for both; for when we divide each magnitude by itself ,
we introduce the inequality again into the meaning of the
condition which was to have been alike for both. But
this explanation will not cover the following third case;
multiply both by o, and the product in both cases alike
is o. It cannot be denied that the operation of taking
a magnitude no times has but one meaning, and does
not as m the former case depend upon the value of the
magnitude to which it is applied : on the other hand it
may be remarked with justice that in this case the meaning
of the like condition or like operation X is precisely of
such a peculiar kind as to annul the inequality of the
magnitudes to which it is applied. Take a fourth case;
if we square the unequal magnitudes a and b> the mean-
ing of the condition to which we subject them is again
dependent upon the magnitudes themselves as in the

Y 2

324 THE FORMS OF PROOF. [Book II.

second case, only with the opposite result; the squares
a 2 and 2 are unequal. Fifthly and lastly the results are
once more equal, for both =i, if we raise a and b to
the o th power. In this case the condition to which we
have subjected the unequal magnitudes a and b seems
to be independent of their value ; but in fact the raising
to the o th power is a quite inconceivable operation; we
must remember that in general a m ~~ n is merely another

a m
expression for ? and that accordingly a l ~~^ ? which is equal

to a Q > is identical with-? and therefore this fifth case is
a

identical with the second. If we wish to avoid all these
ambiguities the only way is to say that of unlike things
under like conditions unlike assertions are true when the
condition is of such a nature as not to affect the unlikeness
of the unlike things : but that like assertions are true
of them when the condition is such as to annul their
unlikeness. But these two propositions are mere barren
tautologies : they do not enable us to decide even so
much as whether the assertions to be made will be like
or unlike without a previous analysis of each case to
teach us what is the general rule M P under which
a and b are really to be subsumed here, and what are
the definite predicates P l , and P 2 which attach to them
in virtue of the special sense in which they, as unlike
kinds of M, partake of this universal P. When we have
found these predicates P l and P 2 we see whether they
are like or unlike; it is not by analogy therefore, but
entirely by subsumption that the conclusion is arrived at.

215. To the third principle, that of like things under
unlike conditions unlike assertions are true, a higher value
may be assigned ; it would in fact be inconsistent with the
law of identity if an identical subject under really different
conditions showed no trace of the influence of this differ-
ence, and I shall have occasion some way further on to

Chap. IV.] UNLIKE CONDITIONS. 325

make use of this proposition as a not unfruitful maxim
in the treatment of philosophical problems. But for the
present what strikes us is the number of apparent ex-
ceptions. How could the engineer solve the problem of
constructing a machine which under changing conditions
regulates itself and maintains a uniform motion, if the same
subject or material substratum under different conditions
absolutely must exhibit different effects ? A closer exami-
nation removes this objection ; it teaches us that in the
cases here concerned either the unlike conditions are not
simple but go in pairs, or that the like subject is not simple,
but a whole of various parts. But two pairs of conditions
may with regard to a definite effect be equivalent, because
the unlikenesses of the several members, in virtue of the
definite relation which subsists between them, annul one
another till the remainders are like ; on the other hand
various unlike conditions may so work upon the various
parts of a whole that the several effects in each case modify
one another till the resulting state of the whole is like. A
simple body which is out of all relation to others can never
receive under the impulse of a force a the same motion
that it receives under the impulse of a force b unequal to a.
But under the simultaneous influence of a and b it may be
moved at the same speed and in the same direction as under
the combined influence of c and d: if these four forces operate
in the same straight line, the equality of their algebraical
sum, i.e. the condition that ad = cd, is enough to give
a like motion to the body; or in more general language,
every motion m may be conceived as the resultant of a
countless number of different pairs of components.

Now this result may be exhibited in various ways. If we
regard the sums a b and c d as the conditions to which
the body is subjected, then the conditions themselves are
like one another, and the case comes under the principle
that of like things under like circumstances like assertions
are true : but if we leave the several forces separate, the

326 THE FORMS OF PROOF. [Book II.

case seems to make an exception to the third principle.
Nevertheless I should like to maintain that this third
principle is universally true ; for its true meaning plainly is
that the sum of all the effects experienced by the same
subject or substratum under different conditions will always
be different. And so even if two pairs of conditions are
equivalent in respect of one kind of effect which they
produce on the same subject, it does not follow that they
are also equivalent in respect of all their effects, and it is
not proper to attend to the former only and neglect that
part of their effect which is unlike. If a and b work upon
a body in opposite directions, and c and d also in opposite
directions, and if their sums or differences a + b and c d
are like, the body certainly experiences the like motion m,
and remains at rest if a b and c-=d- } but it obviously
experiences very different pressures according as it is two
large or two small forces that hold it in equilibrium.
Though a self-compensating machine continues to act alike
under constant and under varying conditions, yet the posi-
tion of its parts changes as the conditions change, and it
wears out faster when it is obliged to exert its compensating
powers than when it leaves them unused, the conditions
remaining uniform. If full sunlight falls upon one scale
of a balance suspended in a vacuum, while the other is in
shadow, the equilibrium is not disturbed, but the first scale
is warmed and expanded more than the other. Lastly if we
multiply a first by a b and then by b a, these conditions are
certainly quite equivalent in respect of the magnitude of the
resulting product, but not in respect of its structure, and
a a b is in any case a different combination from aba. It
would be easy to add to these examples, already sufficiently
various, and thus to confirm the universal truth of the third
principle ; but after all it is but of very little use for a proof
by analogy; it never enables us to establish what all analogy
aims at, viz. that in a second case the same thing happens
as in a first, but only brings us to the negative conclusion,

Chap. IV.] IDENTITY AND SIMILARITY. 327

that any difference of the conditions in the same subject
makes the likeness of the total effect impossible ; what is
still like in this effect, and what unlike, we can never tell
without an enquiry of another kind.

The fourth principle needs but the barest mention ; that
of unlike things under unlike conditions unlike assertions
are true is, after all that has just been said, so evidently
unfounded or ambiguous, that no useful application of such
a statement is conceivable.

I will only add in conclusion that the trains of thought to
which the title of proofs by analogy is supposed to be
appropriate do not even proceed directly from these prin-
ciples, though they must be traced back to them. The
presupposition on which they rest is rather that of similar
things under similar circumstances similar assertions are
true. Now similarity 1 is always a mixture of identity 2 in one
respect and difference in another ; if therefore it is difficult
to base any valid inference upon the foregoing propositions
which separate the mingled elements, it is still less possible
to do so when the two are indiscriminately fused together
in the resemblances to which appeal is made. I think
therefore that I have sufficiently shown that there is no
such thing as a proof by analogy ; though I do not by this
intend to deny that the observation of even remote resem-
blances is of great assistance to the discoverer both in
detecting new truths and in finding grounds for proving
given truths ; for, to sum up my meaning briefly, there is
no need to impugn the abstract validity of these three
principles, but only their fruitfulness for demonstration.
We cannot on the ground of the unanalysed similarity of
two subjects transfer the predicate of one to the other, but

1 ['Aehnlichkeit.']

2 [' Gleichheit/ It is impossible to adhere to a single rendering for
* gleich.' Thus * unlike ' applied to magnitudes as on p. 323 might mean
heterogeneous ; ' ungleich ' is therefore rendered there by * unequal ; '
but in the rest of this passage by * unlike.' Cp. Metaphysic, sect. 19,
note.]

328 THE FORMS OF PROOF. [Book II.

dhly on the ground of their demonstrated identity, identity
at least in respect of the conditions upon which the predi-
cate in question everywhere depends; and this always
brings us back to setting down a universal proposition
M P and subsuming both subjects under the determining
conception M.

216. We have still to consider those mathematical argu-
ments which are commonly called proofs by strict analogy.
As the name analogy originally meant proportion, every
procedure that leads back to proportion has a reasonable
claim to the title ; the effect of common usage however is
such that when we hear of an inference by analogy we
expect an argument which reasons directly from similars to
similars, without needing to take a circuitous route through
a higher universal. But the methods employed by mathe-
maticians cannot be thus opposed to proof by subsumption.
A proportion between four determinate magnitudes, a\bc\d,
is merely the expression of a fact ; it only becomes a source
of fresh inferences when the last two members are left
indeterminate ; but in this form, a : b = m : n, it is the ex-
pression of a universal law ; it asserts that the magnitudes
yielded by the problem now before us at the moment are
connected together in pairs in such a way that in every pair
one member is to the other as a : b. If we give any definite
value to m and n we get a syllogism in Darii, all the pairs
of magnitudes which the problem yields (M) have the ratio
P y viz. the ratio a : b \ but m and n (the S or subject of the
minor premiss) are such a pair ; therefore m and n are to
one another in the ratio a : b. No doubt this reduction to
the first figure is very tedious ; but we deceive ourselves if
we fancy, because of the shortness of the formulated ex-
pression which the nature of the subject-matter makes
possible in mathematics, that the train of thought also in
a simple proportion is something shorter than that here
stated. Even the simplest example of the rule of three
is worked in this way. We say, if i' pound costs two

Chap. IV.] STRICT ANALOGY. 329

thalers, 10 pounds cost 10x2 thalers : here we assume,
what seems to us self-evident, that the ratio between the
quantity of the article and the price is always the same ;
accordingly we take the ratio of the one pound to its price
as a general rule and bring the ratio of the 10 pounds to its
price under it as a particular case of the rule : but the
dealer perhaps sells the 10 pounds for 18 thalers and
thereby shows that what we assumed is not self-evidently
true in all cases, but that we really had to make the
assumption for the purposes of our calculation : further it
is evident that we tacitly conceive m and n as standing for
quantities of the same article and of the same unit of
currency as a and ^, and so in this respect also take the
first case as the general rule and subsume the second case
under it. Every general equation which exhibits one and
the same content under two different forms is equally a
general rule, which holds good only for that kind of magni-
tudes which, by a convention which finds no expression in
the formula itself, we intend to denote by these particular
letters, and for which we originally showed the equation to
be valid. It is not allowable therefore to substitute for
the magnitudes m and n which occur in an equation any
other chance magnitudes M and v, and to regard the equation
as still valid : we must know beforehand that //, and v can
be subsumed under the species m and n of which the
equation has been proved to be true. Suppose we have
proved by actual multiplication and by the argument from
n to n + i that

that does not give us the right to infer also that

(--- 1)

\m / o

X

i.m mi. 2
for in the first formula m stood only for the class of positru

330 THE FORMS OF PROOF. [Bookil.

wLole numbers, for which alone the proof by multiplication
was feasible, and a fraction cannot be subsumed under it.
If on the other hand we had found means to prove in the
first instance that the binomial theorem in the first case holds

true for the fractional exponents ? whatever positive value

may be assigned to m and #, we might have deduced the
first formula directly from this, since every whole number
m may be expressed in the form of an improper fraction.

217. In conclusion I should like once more to connect
what I have said with the dictum de omni et nullo or the law
of disjunction. If S l and S 2 are two species of the genus
M or two particular cases of the universal M^ and if P may
be predicated universally of M^ we know that P may be
predicated of S l and *S 2 not in this universal form but in
the modified forms P 1 and P 2 . Now in a special case it
may happen from the way in which the various predicates
P Q R are connected in M, that the various groups of
characteristics / 1 q x r l , / 2 q* r 2 , ^ q z r z which they form in the
several subjects s l s 2 s^ must be identical with one another ;
they then make so to say a secondary predicate O, which
may be ascribed to M itself, and which equally attaches
without modification to every species of M. Thus the con-
ception of the triangle M requires three angles pqr, but
the various values of these angles in the various kinds of
triangles always make up the same sum II = 2 right angles ;
this identical characteristic IT therefore attaches to all
triangles and we may at once ascribe it to any single
triangle when we have simply subsumed it under its genus.
But apart from such special cases the / 2 or q* that will be
proper to an s* remains indefinite, with the single limitation
that it must be a kind of Q, and that it must always be
present, even though its value diminish to nought, in which
case this nought must be capable of explanation. If this ^ 2
is to be determined, there must be a rule according to which
the specific peculiarity of S l (which makes it not only a kind

Chap. IV.] FROM SPECIES TO SPECIES. 331

of M but this particular kind) helps to determine the modi-
fications of the general characteristics of M^ in this case
the modification of Q ; and we must assume that the peculiar
nature of S' 2 will follow the same rule in determining (f, the
modification of the general characteristic Q which is appro-
priate to it. If we know this rule we can determine ^ 2 , and
this is precisely the case which is called inference by strict
analogy, though as we have seen this rests upon nothing
but the subsumption of a case under the like universal rule.
But when this rule is not known, we still feel inclined to
find out q 1 by considering the resemblances and differences
in the relation of S l and S 2 to each other and to M, and the
procedure based upon this we usually call inference by
analogy; but it only enables us to guess the right result,
never to prove it. It was known by the forty-seventh pro-
position that for right-angled triangles the square on the
hypotenuse h is equal to the sum of the squares on the
sides a and b which enclose the right angle. As this
relation can depend upon nothing but the general pro-
perties of the triangle, the right angle, and the length of
the sides, it is a quite justifiable impulse which bids us
seek an analogous proposition about the square of the
subtending side for other values of the subtended angle.
If we simply put the formula in the general form // 2 =# 2 -}-/> 2
there is t no longer any mention of the right angle \ but the
formula we are seeking must mention the subtended angle,
for it is evident at a glance that, a and b remaining the
same, h gets longer as the angle increases and shorter as it
diminishes. Accordingly to make the Pythagorean formula
complete we must add another term which will become
nought when the included angle </> = 90 : and as we cannot
measure h by the angle itself, but only by a length de-
pendent upon it, or by a numerical coefficient dependent
upon it that determines another length, we may set down
tentatively %'*= a?+ $* m cos (/>. The alternative sign +
is seen at once to be needless when we reflect that when <J>

332 THE FORMS OF PROOF.

increases beyond 90 h still increases but the cosine becomes
negative; we only need the minus sign therefore in the
formula. In order to find m which is as yet indeterminate
we turn to the two limiting values of $, <f> = o and (/> it.
In the latter case Ji l becomes equal to (a -f Vf and cos
<j> = i ; in the former case ti* = (a frf and cos <$> = -j- i ;
both cases alike give us h 2 = # 2 + ^ 2 2 # cos 0. Now
this formula is in fact correct for all values of $, but it is as
yet by no means proved ; it covers with certainty only the

three special values of <, viz. $ ~ , </> = -TT, </> = -, from

which it was obtained : it would be easy to find another
formula, e.g.

/& 2 =0 a + P- 2^COS<|>.COS 2 (<7T- (/>),

which would also cover them; which of the two is also
satisfied by all the intermediate values of <j> remains un-
settled, till by an easy geometrical construction, with the
help of the forty-seventh proposition, we decide that the
formula we first took is universally true. I have dwelt upon
this simple example in order to show how many subsidiary
considerations are necessary before our efforts to discover
new truths by the analogy of given truths can even be put
into a path which promises success.

CHAPTER V.

The discovery of grounds of proof ,

218. IN any demonstration of a given proposition T
the most important thing is to find the major premiss G,
from which by appropriate subsumption T is to follow as
necessary consequence. This problem, obviously a problem
for the discoverer, does not admit of any logical rule by
which the solution could always be found with certainty,
without counting upon the free co-operation of the sagacity
of the individual enquirer. We must suppose that previous
reflexion has already supplied a number of general truths,
which are related to the content of the given T in such a
manner as to be serviceable for the purpose in hand, and
which, recalled to consciousness by the similarity of the
matter in question, suggest themselves to the seeker as
grounds for explaining the given proposition. But over
and above this he must have the keenness of mental
vision which detects among these truths the appropriate
ground of proof, and sees the changes which perhaps are
necessary to the subsumption of the given proposition
under it, and this we must allow is to a large extent matter
of native talent and not even independent of the moods of
the moment. The logical relation however which subsists
between the parts of a true and therefore demonstrable
proposition must be able to give us at any rate such a clue
as may save us from groping entirely in the dark and to
some extent put us into the way of finding, after further

334 THE DISCOVER Y OF GROUNDS OF PROOF. [Book II.

starch of course, the ground of proof. This clue lies in
nothing else than the . fact which we remarked some time
ago that every true universal proposition T, when we sup-
plement and complete its subject and its predicate by all
the subsidiary characteristics which are hinted at or implied
though not expressed, must become an identical proposition.
If then for the conception S, which occurs as subject in the
proposition T^ we substitute this completed sum of the
several ideas which it contains in the forms of combination
proper to them, this must include the ground which justifies
the predicate ; on the other hand if we substitute for P in
its completeness the sum of the several ideas included in it,
this must include all the requirements which the subject
must satisfy in order that the proposition T may be true.
I will attempt to illustrate by a few examples the use of this
clue, and as discovery and proof here in fact follow the
same road, I shall treat some of these examples as proofs
of the given proposition T and others as instances of its
discovery, i. e. of the solution of the question what relation
expressible in a proposition T must subsist between S and P.
219. Suppose first that we have to prove the given pro-
position T, that the angle in a semicircle is a right angle.
By analysis of the subject we find that by the angle in
question we have to understand one whose enclosing lines
start from the extremities a and b of a straight line a h and
intersect each other at a point in the circumference of a
circle described about a b as diameter. Now if the second
part of this definition, which determines the position of the
point of intersection <?, is to be satisfied, the distance of e
from c the point which bisects the straight line a ,
must be equal to half this line, i. e. to a c or c b.
This requirement which follows from the definition of
the subject suggests at once the one slight subsidiary
construction that we need : we must draw this line e t, in
order to bring before our eyes the relations upon which
depends the necessity of the given proposition T. When

Chap.V.] ANALYSIS OF ADJACENT CASE. 335

we have drawn e c the triangle a e b which we already had is
divided into two isosceles triangles a e c and e c b, while the
angle at e is divided into two angles a and /3 : from the fact
that both triangles are isosceles this follows, and so far this
alone, viz. that the angle e a c a and that the angle e b c =
(3 ; but from the way in which these two triangles make up
the triangle a e b, e c being common to both, and a c and c b
falling in the same straight line, it follows further that the
four angles a, a, /3, /3, are together equal to the sum of the
angles of the triangle a e b. We have then 2 (a + fi) = two
right angles, and as a -f ft is the required angle in a semi-
circle, we have found that it is equal to a right angle.

It is not always that we can get what we want by such
an easy analysis as in this very simple case : let us therefore
take another case to illustrate an artifice that is very fre-
quently applicable. We may perhaps already have got a
proposition T which teaches us what is true of a subject
which is not equal to S the subject of the given proposition,
but diverges from it by a difference that can be stated ;
supposing then that by removing this difference we cause
this subject to pass into the given subject S, and are able
to show how the relation expressed by T is altered by this
operation, we shall prove the given proposition T if it is
true, or find the true proposition T if the given proposition
is false, or if none were given at all.*

Suppose the question to be what is the sum of the angles
of a triangle. Assuming that the propositions concerning
parallel lines and their intersection by a straight line have
been established without taking triangles into consideration,
we take two straight lines a d and b c parallel to one another,
and intersected by a third straight line a b in the points
a and b. These three lines thus form no triangle, but an
unclosed space ; but we know S the sum of the two angles
da b and a b <:, and know that it is equal to two right angles.
If we now make the line a d turn about the point a so as to
incline towards b c, there is formed between its new position

336 THE DISCOVER Y OF GROUNDS OF PROOF. [Book II.

and its old one an angle </>, which is taken away from S the
sum of the interior angles ; but at the same time there is
formed between b c and the line which has been deflected
to meet it a new angle, the third angle which together with
the remainder of S the sum of the original angles makes up
the three angles of the triangle now formed, and which by
the propositions about parallels is equal to the angle (/>
which was excluded from S. Thus therefore in the passage
to a triangle from what is not a triangle the sum of the
angles contained by the three lines loses (f> and gains <p ;
it is therefore equal to two right angles in the triangle as
before.

220. Suppose we want to prove or to find the conditions
of equilibrium for a perfectly free and absolutely rigid body,
operated upon at various points by various forces in various
directions. In the conception of a body here employed
perfect freedom needs no further analysis ; as absence of
every conditioning relation to others it is quite clear as it
stands ; only if the relations were present should we have
further to determine their import : the absolute rigidity of
a body means that the distance between any two points in
it is unalterable.

Now if no force were acting upon this body, we should
be able to say of it that it either was at rest, or was con-
tinuing an original motion at a constant speed c : we should
therefore only have to set down c = o in order to express
the conditions of the equilibrium intended, the equilibrium
of rest. But in order to decide how the body maintains
equilibrium when forces are acting upon it we must adopt
the same method as in the preceding case and first see how
it would move if it did move, and then negate all the con-
ditions which would be inseparably bound up with this
motion. This is not merely a useful contrivance without
any logical basis ; for the equilibrium we are now seeking
must be conceived not as mere rest but as the negation of
fhe movements which tend to disturb it. * Now as the only

Chap.V.] CONDITIONS OF EQUILIBRIUM. 337

kinds of motion are motion from place to place, rotatory
motion, and thirdly the combinations of these two, all we
have to do in order to determine the equilibrium of the
body is to consider the conditions of the two first-named
kinds of motion; negate them and the possibility of the
third kind is gone.

221. If we first consider only movement from place to
place or movement of translation, expressly excluding all
rotation, it follows from the definition of rigidity that all the
parts of the rigid body must move onward in rectilinear and
parallel paths and therefore with the same velocity. In
whatever way a force acts therefore, if it has given to a, one
part of the body, a velocity c, it must always, provided the
movement be one of translation and not of rotation, have
given the same velocity to , any other part of the body.
Hence we are able, to our great convenience, in estimating
the movement of translation which finally results from all
the forces acting upon a rigid body to neglect the fact that
they act upon different points : we may treat them all as
acting, in lines parallel to their given directions, at an arbi-
trary point in space, at which we suppose the mass of the
body to be concentrated, and then by the known rules for
the composition of forces determine the resulting movement
jR which they would impart to this point ; the magnitude
and dirg^tion of this resultant R are then identical with the
magnitude and direction of the motion which the body re-
ceives under the united influence of these forces, and it
remains at rest when R = o. If we express this by saying
that the body rests when the effects of all the impulses to
motion which are brought to bear upon it annihilate one
another, the proposition is an identical proposition for
which no reason need be sought : our explanation however
further states the condition under which that annihilation
takes place, viz. the very same condition as that under
which it will take place when all the forces are acting upon
the same point.

LOGIC, VOL- I. Z

338 THE DISCOVERY OF GROUNDS OF PROOF. [Book II.

t

222. In mechanics however it is usual not to state this

condition under this form It = o, but to break it up, for
convenience in applying it to calculations, into three equa-
tions, which I proceed to mention, since the feasibility of a
logical precept is certainly one of the questions which applied
logic ought to consider. If the number n of the forces
acting upon the body be considerable, it becomes laborious
to find the last resultant R by first of all getting a first re-
sultant out of two of these forces, and then a second out of
this and a third force, and so on till the last force is com-
pounded with the last preceding resultant. Moreover the
angles which the direction of each force makes with that of
any other, and which would have to be considered in this
calculation, are seldom included among the data originally
given ; but where these data have to be first determined by
the examination of a given state of things, it will be prefer-
able here as elsewhere to characterise the directions of all
the forces by their relations to a single common standard,
instead of measuring the divergence between every two.
The usual proceeding then is to lay down three axes X YZ,
at right angles to one another, and then to determine the
direction of each force P by the three angles a ft y which it
makes with these axes or with lines parallel to them, at the
same time conceiving each force as resolved into three
components parallel to these axes, which forces will accord-
ing to a familiar proposition be P cos , P cos /3, and P cos y.
The three sums then made by adding together all the com-
ponents of like direction, i. e. the sums 2 P cos a, 2 P cos /3,
2) P cos y, will be the resulting forces which tend to move
the body in directions parallel to the axes X Y and Z re-
spectively : if each of these sums as they stand be equal
to nothing, the body does not move from its place in
any of these three directions, and therefore does not move
at all, for any movement in an intermediate direction
4 would include a simultaneous change of place in the direc-
tion of two of these axes at least, and this has just been

Chap.V.] CONDITIONS OF EQUILIBRIUM. 339

denied. So instead of R = o we have these three equations,
2 P . cos a = o, 5 P . cos j3 = o, and 2 /> . cos y o, to
express the condition which annihilates all movement of
translation.

223. We have still to look for the other conditions which
make the rotation of the body impossible. Suppose now
that a straight line rotates about one of its points ; then
with the exception of this one point which we regard as
fixed (thus making it impossible for the whole line to have
any movement of translation) all the other points of the line
alter their co-ordinates. The line therefore cannot rotate if
two of its points have unalterable co-ordinates. But though
the line be fixed along its whole length, a plane which con-
tains it may rotate about it : then all the points in the plane
which lie outside this axis alter their co-ordinates : the rota-
tion of the plane therefore becomes impossible if any point
in it outside the axis be fixed, or in general if the three
angular points of a triangle drawn anywhere in the plane be
fixed. The same condition is obviously sufficient to make
rotation impossible for a rigid body, every point of which is
at an unalterable distance from every point in a fixed plane
taken at will in it. The condition which prevents rotation
therefore might be expressed by saying that the three an-
gular points of a triangle drawn anywhere within the body
do not alter their co-ordinates. But the proof that this con-
dition was fulfilled would not be at all a convenient one : in
order to prove it by applying the previous three equations
to each of these three points we must be able to prove what
is the resultant effect at each of them of all the forces acting
not at this point but at other points : but this, as will easily
be seen, is the very thing that we are still trying to ascer-
tain. We must take another course therefore, and, since
the position of the triangle just mentioned is perfectly arbi-
trary, the course which most naturally suggests itself is to
dispose its three angular points in the three axes X Y Z 9 by
reference to which we have already determined the directions

Z 2

340 THE DISCOVER Y OF GROUNDS OF PROOF. [Book II.

of all the forces in operation : but the position in each axis
of the angular point which we place in it is also perfectly
arbitrary: we may therefore regard every point in each axis
as a point whose position is unalterable, i.e. we may regard
the three axes themselves as three fixed lines, in relation to
which, if rotation is to be excluded, no point of the body
can change its position and distance. If finally we consider
the three axes as three dimensions which lie within the body
itself, or as identical in position with three series of points
in the body at right angles to one another, it follows from
the definition of rigidity that the fixity in space of these
series of points is all that is required to make any change of
place impossible to the remaining points of the body. The
problem therefore reduces itself to showing that all the forces
in operation are unable to impart a rotatory movement in
any direction to any of these three series of points, or to
any of the three axes X YZ now conceived as capable of
moving out of their previous direction.

224. This last way of treating the matter however would
not serve as a convenient basis for calculation except when
the directions of all the forces concerned passed through the
three axes. This will not generally be the case : in order
to take into account those forces which when produced go
past those series of points without cutting them, we must
substitute for the three lines three planes intersecting each
other at right angles, each of which will therefore include
two of these axes : the direction of each force produced if
necessary must cut one of these planes. The problem now
is to show that all the forces in conjunction are unable to
cause either the planes X Fand X Z to rotate about X y or
the planes Y Z and YX to rotate about F, or the plane
Z Y and Z X to rotate about Z. Let us consider the con-
ditions of rotation about Z. Any force P acting in any
direction upon a point of the body whose co-ordinates are
x y z, and making with the three axes the angles a ft y, can
as before be decomposed into three forces P cos a, P cos /3,

Chap.V.] CONDITIONS OF EQUILIBRIUM. 341

P cos y, parallel to the three axes. The last of the three?we
need not consider here ; it could only cause a movement of
translation in the direction of the axis Z, which is already
excluded Jby the equations of 222, or a rotation of the
plane X Y about Xor Y, which also need not be considered
at present. Of the two other forces P cos a is perpendicular
to the plane Z Fand Pcos (3 to the plane ZX; the two
tend, as is shown by an easy construction, to cause the
planes ZJfand Z F, and so the body in which these two
planes are immoveably united, to rotate in opposite direc-
tions : the direction of the rotation which actually results
would therefore depend upon the difference between the
two forces. Not simply upon their difference however, for
a proposition which at present we will only allude to, teaches
us that the rotatory effect of a force which is perpendicular
to a line is to be measured by the product of its intensity
into the distance of its point of application from the axis of
rotation. For the force P cos a this distance is j/, and for
the force P cos fi it is x : the difference of the products
y P cos a and x P cos /3, or the difference between the two
momenta, must be equal to nought if P is to cause no
rotation about the axis Z. We must repeat the same con-
siderations with regard to all the forces concerned, and we
finally get, as the condition which prevents all rotation about
the axis Z, the equation

S ( y P cos a x P cos /3)= o.

The other equations which make rotation about the axes X
and F impossible, will obviously, as the three directions are
perfectly homogeneous, be of the same form ; and, since
even artificial aids to memory are not beyond the province
of applied logic, I will remark that the equation for non-
rotation about an axis never contains the elements which
refer to this axis, but consists of the sum of the differences
of two products, each of which unites a component force in
the direction of the second axis with that co-ordinate of its
point of application which is parallel to the third axis. The

342 THE DISCOVERY OF GROUNDS OF PROOF. [Book II.

foimula 2 (z P cos /3 y P cos y) = o annihilates rotation
about X ; the third formula 2 (# P cos y 2 P cos a) = o
annihilates rotation about the axis K

225. The proposition about the equilibrium of rotatory
forces which we made use of in the preceding discussion is
easily arrived at in the domain of statics by a slight device
which reduces the question to the composition of motions.
If I now select another mode of proof, I do so of course
with no idea of improving the science of statics ; I only
adopt a treatment which is as far as possible independent
of all merely happy contrivances, in order to illustrate the
way in which the grounds of proof are brought to light by
the analysis of the problem itself. If the rigid line a ,
whose length we will call , rotates about its fixed extremity
a, this implies that all its points describes a segment of a
circle p <o with the same angle co and with a radius p, which
for each point is equal to its distance from the point a. If
now a force W acts upon the point , and causes , in
whatever way, in the time t to pass through the segment
n co, it must likewise have compelled any other point in the
line at the distance p to describe in the same time / the
segment p &> : and conversely, any force which applied at
the point p has caused this point to move through the small
segment p o>, has necessarily compelled all the other points
in the line to describe segments corresponding to their
distance from a. We now ask what must be the nature of
the two forces P and Q in order that when they are brought
to bear at the points / and q respectively they may produce
precisely equal results, and accordingly when acting in
opposite directions upon the line a b may prevent each
other from making it rotate. Now the conception of
rigidity, i.e. the conception of the simple immobility of a,
is too far removed from conceptions of movements, to tell
us how the latter would be affected by the former : we
should have first to conceive rigidity itself as the result of
movements, in order to make it homogeneous with the

Chap.V.] CONDITIONS OF EQUILIBRIUM. 343

other movements upon which it is to exercise a restraining
influence. Further it is impossible to compare P and Q so
long as they act under different circumstances whose
modifying power is yet unknown : we can only estimate
them by velocities (f> and \|/ which they would impart under
perfectly similar conditions to a perfectly similar moveable
object : and lastly though P and Q may be applied at the
single points p and ^, they cannot operate upon them
alone ; in order to set up or to hinder a rotation, the effect
of each must spend itself over all the points in the line a b,
and we must know the mode of this distribution before we
can understand how the effect of the one can annihilate the
simultaneous effect of the other at every point in the line.

226. These requirements we may satisfy in the following
way. Suppose that a b, which is equal to #, is first of all
a perfectly free rigid line, consisting of an infinite number
n of homogeneous points which are compelled (how does
not concern us) to maintain unchangeable distances from
each other. Suppose that a number n of equal and
parallel forces operate perpendicularly upon this line so as
to give to each element of it the velocity oj ; then the total
force W) equal to n o>, will urge the whole line forward, all
the points moving in parallel directions. This movement
of translation passes into a rotatory movement when we
give to the various points of the line various counter-
velocities, which must be conceived as at right angles to a b
not only at the beginning of the rotation but at every
subsequent moment. To the extremity a we assign a
counter-velocity to by which it becomes the fixed point
which our problem requires; to the point b we give a
counter-velocity which = o, so that it maintains undimi-
nished the velocity co imparted to it by W\ the intermediate
points must meet so much resistance as will leave to each
point p, whose distance from the fixed point is /), a residual

velocity whose amount is already known, viz. the arc ~.o),

344 THE DISCOVER Y OF GROUNDS OF PROOF. [Book II.

whose length is to co, the path of the free end, as p is to n :
the sum of the velocities of all the points />, from p = o to

p = #, must be equal to Now a force P, which would
give to a free element the velocity <p, would give to an
element / in our rigid line the velocity - <, if / were sub-
ject to the above-mentioned resistance but able to move by
itself; but as it cannot move by itself, the impulse imparted
to it must distribute itself over the whole line. However
this distribution may be effected, we already know the
result ; it can produce nothing but a rotation of the whole
line, in which every point p receives a velocity proportionate
to its distance from the fixed point and the sum of all the

velocities is - Every point p therefore receives the
2 n J r

velocity - - - Precisely similar statements may be

made about a second force Q, which would give to a free
element the velocity \ff, but to an element q of the line

which is fixed at one end would give the velocity ~-^\
when applied at q it would give to any other element p of

the line the velocity - - - - Now if these two forces
3 n \ji n\

operating at p and q or the two velocities produced by
them are to be such that when acting in the same direction
either would annihilate one and the same third movement
of the line, or that when acting in opposite directions they
would counterbalance each other, then for any point p the
two expressions which we have just found for their effects
must be equal to one another, and so therefore / <t> = q ty,
and </> : ty = q :/. In other words the length of leverage
must vary inversely as the strength of the force.

227. The following would be a very plausible, and yet
an inadmissible way of deducing the same proposition.

Chap.v.] CONDITIONS OF EQUILIBRIUM. 345

Suppose that at the same point m of a lever playing in a
vertical plane two equal forces P and Q are acting in
opposite directions; it is self-evident that under these
conditions equilibrium will be the result. Now if, as is
commonly done, we imagine Q to be a weight, suspended
by a hook or cord at m, and P as a strain exerted from
above, we tacitly assume that it is indifferent~whether of the
infinite number of infinitely thin perpendicular strips
into which Q may be decomposed in thought each severally
grapples the point of the lever which it would touch if
produced, or whether all these several forces operate upon
the lever only through a single representative which unites
them all, viz. the cord. Once assume this, and it must
also be indifferent whether we conceive Q as one body, or
as divided perpendicularly by a geometrical plane into two
halves which touch one another at the surface of section,
and each of which is attached to the lever by a separate
cord which unites all its forces in one resultant. If then m
was the distance from the fulcrum of the original point of
attachment, m x and m + x are the corresponding
distances of the new points of attachment of these two
cords. In other words equilibrium is preserved when two

forces each of which is equal to - > and whose sum = P,

are applied at equal distances right and left from the attach-
ment ol the opposite force/*: for the cords themselves, or their
tensions, are now the forces which are directly applied. Now
so long as these tensions are the resultants of the forces of

gravity united in the two bodies ? it is evident that it is

quite indifferent how these bodies ~ are shaped in other

respects, indifferent therefore whether they still touch one
another as before, or whether by increase of their length
and diminution of their thickness they become two separate
bodies with a space between them. If we follow out this

346 THE DISCOVERY OF GROUNDS OF PROOF. [Book II.

line of thought we see that it is quite possible to carry the
displacement of one to the left and of the other to the

r 2

right by equal distances x as far as we please, till at last x
becomes equal to m : when that is done one > say the one

that was displaced to the left, has reached the fulcrum a,
and no longer produces any effect upon the lever : the

other has arrived at the distance 2 m from the fulcrum,
2

and the equilibrium is still preserved under the condition
.that P, which = Q, operates at the distance m from the

fulcrum, while operates at the distance 2 m.
2

But though this exposition brings the matter before us
very plainly, it is nevertheless absolutely inconclusive. So

long as x was less than m^ the that was moved away to

the left had still a recognisable and intelligible influence
upon the equilibrium of the lever; we could still see
plainly that it together with the other half that was moving
away in the opposite direction made up the force that was
sufficient to counteract P\ but so soon as x becomes equal

to m, and the effect of this altogether ceases, there is a

break in the thought : for one of the points of relation has
vanished, and our whole reasoning was founded upon its
relation to the other. For when we first applied Q at the
point m itself, and then disposed the two halves of Q
symmetrically on either side of m, what we inferred held
good in the first instance for the free line a b, which was
supported at m by the force P : the fixing of the end a was
not contemplated at all : though of course the same infer-
ences held good also for the case when a was fixed, so
long as it could be proved that, irrespective of this, equili-
brium was maintained by the way in which the weights

Chap.V.] ROTATION PLUS TRANSLATION. 347

were distributed ; for if equilibrium was maintained thus vt
could not be disturbed by the fact that a was over and
above this regarded as fixed. But so soon as the influence
of one half of Q vanishes, we no longer have equilibrium
on the same grounds as before, and it is by no means self-
evident that the vanished condition is exactly replaced by
the fixing of the end a. We should in fact need for this
special case to find a subsidiary proof which should show
that a being fixed the effect of the half of Q was all along
getting less and less as it approached a, and that equilibrium
was nevertheless maintained; therefore it would continue
to be preserved when the influence of this weight was
reduced to nothing, while the other was removed to a
corresponding distance. But if we examine it, we see that
this subsidiary proof would in reality be the proof of the
main question, i. e. it would be the proof of nothing less
than the proposition that the power of equal forces to move
a lever varies as their length of leverage. This mode of
statement therefore, however plainly it brought the propo-
sition in question before us, did not in the least prove it,
but only assumed it in a circle which it is easier to recog-
nise than to state briefly.

228. Complicated mechanical problems cannot always be
solved by directly compounding all the forces in operation
so as to arrive at their final resultant ; we often have to state
certain universal conditions which it must satisfy, or certain
limits within which it must keep : with these assumptions
then the several data of the given case supply means for the
complete determination of the result. These methods,
among which we need only mention the application of
d'Alembert's principle, are quite invaluable and cannot be
dispensed with : but as they do not clearly show the history
of the result which we calculate by them, we still feel a wish
to employ direct constructions so far as possible. I will
mention in connexion with the preceding problem of the
equilibrium of rotatory forces that of the motion which they

348 THE DISCOVER Y OF GROUNDS OF PROOF. [Book II.

generate when they are not counteracted. The rule for
calculating it is reduced to these two very simple proposi-
tions : (i) if a force acts upon a body that is able to move
freely, its centre of gravity takes the same rectilineal motion
which the whole mass of the body would take if it were
concentrated at the centre of gravity and there acted upon
by the force : (2) at the same time the body takes the same
rotatory motion which it would receive from the same force
if its centre of gravity were fixed. Now in this very neat
division of the result there lies a paradox. For if the
direction of the force passes through the centre of gravity,
there arises according to the second proposition no rotation,
but only a rectilineal movement of translation, and yet we
should suppose that in this case the force was acting upon
the body under the most favourable conditions : but if the
direction does not pass through the centre of gravity, in
which case the force would seem to act under less favour-
able conditions, there follows not only the entire previous
result but also a rotation, which strikes us as an addition
without any obvious reason. If the compound velocities of
the various parts of a body which is at once moving on-
wards and rotating be decomposed into velocities in the
direction of its rectilineal course and velocities in the
directions perpendicular to this and to the axis of rotation,
the sum of all the former components, each multiplied into
its differential-mass, is equal to the product of the whole
mass multiplied into its rectilineal velocity ; and we easily
convince ourselves that when the body is at once rotating
and advancing, though the several elements have various
velocities in the direction of its course, yet the sum of all
these velocities is neither increased nor diminished, but
only otherwise distributed than it would be in the same
mass advancing without rotating. But the other com-
ponents remain, and though they have opposite signs for the
two halves of the rotating body, yet they do not on that
account annihilate each other: they are motions which

Chap.V.] ROTATION PLUS TRANSLATION. 349

actually occur, and we are forced to ask where they comr
from.

229. It is sufficient to answer this question in the
simplest conceivable case. Let a and b be two equal
masses, which we conceive to be concentrated at their
centres of gravity : suppose that they act upon each other
so as to. remain always at the same distance a b from one
another : we may say then that a and b are united by a
rigid unchangeable line a b which has no mass. In order
to simplify the figure to be drawn, conceive a b to be so
fitted into the angle of two rectilineal axes which intersect at
O that a lies upon the axis X and b upon the axis Y: at
starting then we have, for the mass <?, x = O a and_y = o>
and for , x = o andj/ = O , while for the centre of gravity
of the system a -f , which lies in the centre of the line a b t

we have x = andj/ = We will now suppose that

a certain velocity is imparted to the mass a in the direction of
the axis X, and that a a is the path which it would traverse in
an indivisible moment of time under this impulse if it were
free. As no force is acting directly upon the mass , it would
then remain at rest, and the line a b which expresses its
distance from a which has moved away would be longer
than the original line ab. But the forces in operation
between a and b, which according to our assumption main-
tain the distance a b unaltered, oppose themselves at every
moment to the beginning of this elongation the measure of
which would be a b - a b, and prevent it, by making the
two bodies approach one another in the direction of the line
at the extremities of which they would be found if the
elongation actually took place. Since neither of the two
masses can one-sidedly compel the other to follow it, but
both masses, being assumed to be equal, must by the
principle of the equality of action and reaction displace each
other to the same extent, we shall find their new positions
a 1 and fi by cutting off from the line a b the length a 1 equ4

350 THE DISCOVER Y OF GROUNDS OF PROOF. [Book II.

{ o , and from the line b a the length b /3 also equal

to If from a 1 we let fall an ordinate, which we

2

will call dy y upon the axis X, and from /3 let fall a perpen-
dicular, which we will call dx^ upon the axis Y, we have two
equal and similar triangles, and thus we get for a. 1 and /3,
the two extremities of the now displaced line a l>, the
ordinates dy and O b dy respectively ; and therefore for
the centre of gravity, which is still the centre of this line, we

have y = : but this was also the ordinate of the centre
2

of gravity before any velocity was imparted to it : the centre
of gravity therefore has received an impulse to move in a
direction parallel to the axis of X, i.e. in the same direction
in which a would have been impelled to move if the force
had been brought to bear directly upon it. At the same
time we have for the extremities a 1 and /3 the abscissae
Oa + aa d x and d x respectively, and thus for the new
position of the centre of gravity we have the abscissa

; therefore, since the abscissa of its original

2

position was it has received half of the velocity a a

which the force applied to a tended to impart to <z, and this
is precisely the velocity which the same force would have
imparted to the whole mass of the system (which is a f b
or 2 a) if that mass had been concentrated at the centre of
gravity and the force applied to it there.

These considerations apply to the first instant of the
whole motion, in which (as is usually assumed) the force
applied to <z, working instantaneously, gave ' it a certain
velocity without any lapse of time, and in which the
corrective reaction of the forces at work between a and b
also took place without lapse of time. Since from this
instant no external force any longer " operates, all the

Chap. V.] /? TA TION PL US TRANSLA TION. 3 5 1

motions produced will simply continue according to the lafr
of persistence, only the internal forces that act between a
and b have to be continually at work in order to prevent a
and b from flying off at a tangent, and to maintain them at
a constant distance from their centre of gravity ; they thus
generate a rotation which is circular in relation to this point,
and since they are continually diverting the two masses
from their momentary direction into another without any
breach of continuity, the rotation takes place uniformly in
a circle and with the same constant velocity with which
both masses are impelled in a straight line at the first
moment.

Lastly if we move back a 1 /3, keeping it parallel with itself,
till its centre of gravity coincides with that of a , the two
lines will make with one another at the centre of gravity an
angle $ equal to that which a b would make with a b at
the point b if b were a fixed centre of rotation and the
external force had only had to move the mass a under the
condition that it should always be at the same distance a b
from b. The length of the curve which a would then have
described would have been a b . </> ; the length of the curve
actually described by a in rotating about the centre of

gravity which we regard as fixed is \ and this is pre-
cisely tke velocity which the force must impart when it has
at the same time to move the mass b in the contrary
direction. From this we see that a momentary external
force, whether its direction pass through the centre of
gravity or not, always produces in the body the same sum
of movements of translation : the rotation which is added
in the second case is due to the internal forces which act
between the parts of the system moved. But these forces
are by no means inoperative even in the first case where no
rotation occurs : but in the first case their only effect is to
cause the several parts of the mass, which are arranged in a
straight line at right angles to the direction of the motion

352 THE DISCO VER Y OF GROUNDS OF PROOF. [Book II.

imparted, to maintain this order during the onward move-
ment, an effect which reveals itself in no relative movement
of the parts about their advancing centre of gravity so long
as we proceed upon the assumption that the body is
absolutely rigid ; but it would at once announce itself in
such movements if we conceived say three equal masses abc
united to one another by pliable cords and then imagined
an impulse to be brought to bear upon the centre of gravity
of the whole system which lies in b.

230. In the analysis which is required for the discovery
of the grounds of proof we try not only to bring out the
elements which are essential to the truth of the consequence
to be proved, but also to eliminate those that are unessential
for that purpose. For instance it is not uncommon in
answering statical and mechanical questions to start from
the supposition of a rigid line without mass. Now it may
be granted that in the conception of a finite straight line
the characteristic of finiteness implies the constant contact
of each point with two neighbouring points, and the straight-
ness implies that the line is rigid and cannot bend : only as
a mere geometrical line it is not an object that could be set
in motion by forces at all ; the capacity of being affected by
forces belongs to the lineally arranged mass only, and it is
only the forces exerted upon one another by the minute
components of the mass that actually give to this material
line the rigidity and unalterable length which is merely
demanded in the geometrical conception.

A line without mass therefore is not a happy expression,
and does not in fact convey that which we really mean and
upon which we build in carrying out such enquiries. A line
must undoubtedly have mass if forces are to cause it to
rotate about its extremity, but with a view to the laws which
regulate the effect of these forces it is only necessary that
the mass be the same at any cross-section of this material
line ; any irregularity in its distribution would constitute
a special case, in determining which we* should have to

Chap. V.] ILLUSTRATIONS FROM MECHANICS. 353

apply with reference to these special data the laws of thgit
simplest case when we have the problem in its purest form ;
on the other hand it is perfectly indifferent for these laws
how great this mass is ; the proportions between the forces
and the leverages necessary for equilibrium are precisely
the same whether the lever be thick or thin, whether its
specific gravity be greater or less. When we speak of a line
without mass therefore we do not strictly speaking set down
its mass as nothing but rather as a unit, and further as a
unit to which any value great or small may be given, and
which disappears from our further calculations just because
as an equal factor of all the terms that stand in proportion
to one another it does not in the least contribute to deter-
mine or to alter the relation which subsists between them.
This was the thought upon which the foregoing exposition
rested. The line a b was conceived as a line of mass, and
every one of its points as a differential of the mass : it was
only this that made it possible to speak at all of a force W
acting upon a <, and to set down this force W as equal to
n co, equal to a sum of individual forces each of which was
such as to give the velocity co to the differential of the mass.
But we should have gained nothing by constantly taking
count of the mass in our calculation ; only the value of o>
would have come out differently according as the mass of
the line or of every one of the n parts of it which we dis-
tinguished was conceived as greater or smaller ; the relations
between P and Q would have undergone no change so long
as both were always related to the same mass. The division
of the labour of proof therefore which is here introduced
does not consist in first putting mass altogether out of sight
and proving the law in question for the line without mass,
and then enquiring in the second place what becomes of
this law when mass is given to the line ; on the contrary we
took count of this mass at the first step, but found that its
magnitude has no influence upon the general form of the
law: upon this ground then we may proceed in a second

LOGIC, VOL. I. A a

354 THE DISCOVER Y OF GROUNDS OF PROOF. [Book II.

enquiry to ask how differences in the magnitude and
distribution of the mass affect the absolute values of the
magnitudes which are to be determined by the law. As
soon as we take this line without mass literally and think of
its being moved, we become involved in absurdities through
which we can never fairly make our way, since the combina-
tion of ideas upon which they rest is in itself an impossible
one. What is supposed to happen when one extremity b of
such a line receives a velocity c ? It cannot separate itself
from the rest of the line, for then it would not be the line,
but only the free point b that was moved : but as the line
has received no motion how can it follow the point? It
may perhaps be supposed that this line would rotate : then
the point b would have to communicate its velocity to the
other points, and that in degrees, more to the nearer and
less to the remoter points; but we cannot see how this is
to be measured, for all the forces are absent here which
operating between the minute parts of a mass might cause
the impulse received by one part to extend itself to the rest
of the series, so that every member of it might at every
moment receive a definite proportion of the impulse.
Finally as there is here no reason for such an apportion-
ment of the effect we might instead of this come to regard
the whole line a b as a unity so closely bound together that
every part of it, separable only to our thought or sense,
immediately assumes the same states that are set up in
any other part : setting aside the question whether every
part of the line would then receive the whole velocity c or

only - j the result would at all events be that the line a b

remains at rest when b receives the velocity c and the other
extremity a receives an equal velocity c. All these
absurdities are avoided by the admission that only a line
that has mass can be moved, not a line that has no mass.

231. In the subsidiary processes also, the substitutions
and transformations by which we endeavour to make the

Chap.V.] ILLUSTRATIONS FROM MECHANICS. 355

given circumstances accessible to our judgment, we have to
avoid suppositions to which, however much they may help
the imagination, no real meaning can be given. To illustrate
this I will mention a proof which is often employed to
demonstrate the parallelogram of forces. The body is
supposed to move in a plane from a to c, and at the same
time this plane is supposed to move from a to b ; and in
this way it is fancied that the course of the body from a to
the end of the diagonal of the parallelogram abed has been
ascertained. This involves two assumptions which are not
expressed but to which expression must be given ; they are
first the assumption that the motion of the plane will not
interfere with the motion of the point in the line a t, and
secondly that the moving plane will carry with it the whole
line a c together with the body. Now an empty surface in
motion is sufficiently for removed from anything that can
actually occur, but it is still harder to understand how a
body can stick to it while it moves. And yet it is very
necessary that it should so stick : for if the body be upon a
very smooth table and we give it a push towards a r, giving
the table at the same time a push towards a b, the body will
not go with the table but will part company while the table
flies away from under it. But if we supply this necessary
condition, i.e. if we say that the body continues to move
undisturbed towards c, while a c at the same time is com-
pelled to move towards b and to take the body with it, the
whole proposition becomes an empty tautology, and that
which is assumed is precisely that which was to be proved.
It must rank then only as one of the means which may be
employed to give us a picture of an already demonstrated
truth.

232. Among the numerous other proofs of the same
proposition several proceed from a common starting-point
which is of interest for the logician. They begin with a
statement of the special case in which two equal forces a
and b impel the body in two directions, and it is regarded

A a 2

356 THE DISCOVER Y OF GROUNDS OF PROOF. [Book II.

as self-evident that the direction of the resulting motion will
bisect the angle between these two directions. But this
assumption includes the further assumption that if the forces
be unequal the resultant will divide the angle into two
unequal parts, and since it is impossible that the kind
of this inequality should be independent of the relation
between the magnitudes of the forces, seeing that the fact
of the inequality depends upon it, this assumption rests
on a more general assumption, viz. that if two conditions
a and b tend to give each a different form to a result c, the
recognisable influence of the two in the actual form of the
result will be proportional to their magnitudes; if then
a and b are equal, c will be as far removed from the result
which would have followed from a alone as from that which
b alone would produce. Now I cannot see why we should
appeal to this proposition once only when we are intro-
ducing the proof, and then conduct the proof itself by other
complicated considerations : whatever be the forces a and b
and the degree of their inequality, we may say universally
that the extent to which the moved point is deflected by
the force a from the path of the force <, and by b from the
path of a, must vary directly as the diverting forces. In
order to turn this logical proposition to mathematical use
we should need first to determine how the two deflections
are to be measured. The nature of the question does not
invite us to apply the ordinary method and to let fall
perpendiculars from the direction of the several paths upon
the resultant or from the latter upon the former : all three
paths are considered not as empty directions in space, but
only as loci which would include the successive situations
of the moved point.

The following treatment is the only one suggested by this
last remark. Let a and /3 be the two points in the paths of
a and b respectively which the moved body would have
reached in the same time / if it had followed the force a
only or b only, and let /> be the point in the resultant at

Chap.V,] ILLUSTRATIONS FROM MECHANICS. 357

which the body arrives in the same time t under the com-
bined influence of a and b\ then p a represents the deflection
from the path a effected by the force , and p /3 the deflection
from the path b by the force #, and p a : p ft = b : a. Since we
can only estimate the magnitude of the forces a and b by
the space which they cause to be traversed in the unit of
time, the ratio a : b is also for the unit of time the ratio of
the spaces traversed in the direction of a and b respectively ;
but it must also have this meaning for any time / and for
any part of t ; for since a and b are regarded as forces that
operate for a moment only, the movement in the direction
of the resultant must take place with constant velocity and
in a straight line : the length which is traversed in the
direction of the resultant therefore will always be propor-
tional to the space traversed in the directions of a and b
within an equal time /, and the lines p a and p f3 which re-
present the deflections will form the third sides of triangles
whose two other sides increase in the same constant ratio.

233. But this proportion tells us nothing about the abso-
lute magnitude of p a and p /3 ; they satisfy the proportion
so long as they are m b and m a ; the value of this m would
still have to be ascertained. Now there is nothing in all
the data of the problem that can help us to determine this :
none of them could have any influence upon it except the
magnitude of a and <, including the ratio of a to ^, and the
size of the included angle; but the suppositions already
made seem to .have taken full count of the influence of these
elements ; and it is quite impossible that anything outside
the data of the problem can contain the grounds of some-
thing that is to flow directly from the problem itself. In
cases of this kind the logical course must always be to
search for the most probable supposition that satisfies the
requirements. The meaning to be attached to this ex-
pression would be very hard to define in general language ;
and my sole purpose in treating this problem is to make up
by an illustration for the want of a precise determination of

358 THE DISCOVER Y OF GROUNDS OF PROOF. [Book il.

the general conception. The most probable supposition
will set down that which in virtue of its nature or magnitude
is the minimum that makes possible the relation which we
know must subsist, and which, if it were to subsist under
other conditions or with other subsidiary characteristics
than those we take, would necessarily furnish special reasons
for inferring them, which reasons are here absent. In the
case before us the proportion p a : p /3 = b : a must always
subsist ; therefore m cannot be nought ; but in order that
it may subsist it is enough to set down m as equal to i ;
and this value of ;// may be regarded as by its nature the
minimum that satisfies the requirements; for any greater
or smaller value, as m = 2 or /// = J, may be treated as
m . i, i. e. as so many repetitions of the unit with the
vanishing of which m itself vanishes and with it the whole
relation. Unity is the only value of m which affirms that
the required relation actually subsists in such a way as to
enable the other special values of m to be effectively in-
troduced as further specific characteristics, in case there be
any reason in the nature of the content under investigation
for preferring one of these values rather than another. Where
as here there is no such reason we fall back upon the
supposition that m = i, a supposition which in any case is
necessary, and therefore is the most probable supposition ;
for under all circumstances, even if m had some other value,
it would hold good at the same time with that value and
equally satisfy the required proportion. Let us then make
the assumption and construct the figure accordingly ; i. e.
let us from a, the extremity of the path traversed in the
time t in the direction of #, describe a circle with radius
equal to the path traversed in the same time towards <, and
from /3 describe a circle with radius equal to the distance
traversed towards a ; then these circles will cut one another
in the diagonal of the parallelogram formed by a and b, and
the direction and length of the resultant are both deter-
mined at once.

Chap.V.] ILLUSTRATIONS FROM MECHANICS. 359

234. But even when analysis has failed to detect any
grounds in the data of a problem for any other than this
most probable supposition, it is seldom possible to be
absolutely certain that such grounds are not there, and
might not be revealed by a more careful analysis. And so
no pains must be spared either to confirm the supposition
adopted by subsidiary proofs upon a different line, or to
establish it indirectly, i. e. to exclude all other suppositions
by showing the contradictions in which they involve us.
We will take this further step then.

It seems self-evident that the resultant of two forces can
never be greater than their sum ; it attains this maximum
when they both act upon the body in the same direction,
and when the included angle therefore is nothing. It has
been objected to this proposition also that it is after all not
self-evident that when a second motion b is joined to a
motion a in the same direction b is simply added to a ;
for it is conceivable that the nature of motion or that of
the bodies subject to it involves conditions which might
even in this case make the resultant greater or less than the
sum of the two. This objection seems to me unfounded,
especially as applied to the case before us. In the first
place when two motions in the same direction are given at
the same time to one body, we may continue to regard
them as two separate motions, but it is only because we
choose so to regard them. They were two motions outside
the body : they may have been imparted to it for instance
by two other different bodies. It may be also that in the
physical act of transmission from one body to another the
motions may lose or gain something : but we are here
speaking not of the mode of transmission, but of the
velocities, so far as they already have been transmitted
to the body in question. In this body, here considered
simply as something moveable, without regard to all its
other peculiar properties, the two do not need to be com-
bined into one, but they are absolutely one from the be-

360 THE DISCOVER Y OF GROUNDS OF PROOF. [Book II.

gcnning, and the resulting velocity is the sum of the two as
surely as any velocity is what it is. But suppose the body
already has a motion a when the second b supervenes ; this
could not make any difference unless the body violated the
law of persistence and altered its motion every instant : for
if it does not alter its motion, i. e. if at the time / it is in
precisely the same condition as at the time /, the motion
b which supervenes later must combine with the still sub-
sisting motion a just as it would have done at the time /
if both had begun together. We may regard it as estab-
lished then that the resultant R of the two forces a and b
acting in the same direction can only be a -f b. Of course
this does not directly help us to estimate the result of forces
whose directions diverge and make an angle <. Meantime
however it is at all events evident that the resultant cannot
increase with the divergence; for then it would be least
when the directions are the same, whereas we have just
seen that it is greatest then, and greatest when they are
opposite, whereas it is evident that it is least then. But it
is equally impossible that it can be independent of the
magnitude of the angle <f> ; and so it must necessarily
diminish as (f> increases, and we may now say that for forces
of any direction the resultant R is either equal to or less
than a -f b.

This conclusion again which is still indefinite may be
brought within narrower limits, if we apply the important
general principle, that objective conditions are independent
of variations in our cognitive procedure. When various
momentary forces to any number we please are brought to
bear at the same time upon a moveable point, the total
result which actually arises can only be one, and therefore
cannot alter with the various arbitrarily chosen series in
which we in our minds first arrange the simultaneous
conditions by pairs, and then again combine the several
results thus obtained. It must be the same in the end
therefore whether we first get the resultant R out of a and b

Chap.V.] ILLUSTRATIONS FROM MECHANICS. 361

and then try to get a second resultant out of R and a, 9r
whether we combine a, b, and a so that, a and a ob-
viously cancelling each other, b is left as this second resultant.
The conception of R therefore as the resultant of a and b
implies that if we again take as components R and a with
its original direction reversed and calculate their resultant
by the^same law by which we get R from a and b, we must
come back to b ; and so R and b combined will bring us
back to a. And this consideration holds good universally,
and quite independently of the still unknown law which
regulates the dependence of the magnitude and direction
of the resultant upon the magnitude of the component
forces and the included angle. From this then it follows
that each of the three forces or motions a, b, R is under the
circumstances stated above the resultant of the other two,
that each is therefore less than or at most equal to the sum
of the other two ; whence it follows that the three jnay be
combined in a triangle, which contracts itself into a straight
line only in the limiting case where one is equal to the other
two.

But as thus obtained this familiar proposition only ex-
presses a relation between the lengths of #, <, and R ; we
must also make out the relations between the angles for
which this relation holds between the sides. If a and b and
the included angle $ be given, the length of R, as yet un-
known, is completely determined ; for these given elements
therefore there is only one possible triangle to be made out
of a b and R. Conversely, given a triangle with a b and R
for sides, there is but one angle <j> which the forces a and b
can make so that R shall be the length of their resultant.
Geometrically R in the triangle increases, if a and b are
constant, as the opposite angle p increases ; mechanically,
as the resultant of a and b y R diminishes as the angle <j> in-
creases ; between the angle p in the triangle therefore and <f>
the angle at which the forces diverge from one another there
must subsist some definite relation which we want to as$er-

362 THE DISCOVER Y OF GROUNDS OF PROOF. [Book II.

tarn. In the triangle made up of a b and R^ R has not the
position which it must assume when it represents the re-
sultant ; in the latter case all three lines must start from a
common vertex A, and it may be taken as self-evident that
R must lie in the angle between a and b. Let us suppose
then that a and b are two forces, as yet indefinite in magni-
tude, put together so as to make any angle $ ; and that R
the resultant, also as yet arbitrary in length, divides this
angle into any two parts, C being its other extremity. Now
as the mechanical relations of which we are here in search
must be independent of the absolute position of the lines in
space, we may first shift the whole system of the three lines
a b and R so that the vertex A falls upon C, and then turn
it, in the plane in which it lies, about C so that the forces a
and b, which in their new position may be denoted by a 1
and l , proceed from C in directions parallel but opposite
to their former directions. Then evidently the resultant R 1
of these forces a 1 and b 1 must be both in position and magni-
tude identical with R, only opposite in direction. Thus
then the direction of the resultant is determined ; it must
be the diagonal of a parallelogram formed by the intersec-
tion of the forces a and b 1 on the one side and the forces b
and a 1 on the other, or by their meeting in a common ex-
tremity, or by their being produced to such an extremity.
But if the lengths of a and b are given, the length of R is
also determined, it must be the third side of a triangle whose
other sides are a and l , which = ^, or b and a\ which = a ;
it is therefore the diagonal of the parallelogram formed by
the lengths of the forces themselves. The figure then shows
that the angle p subtended by R in either of these triangles
is the supplement of the angle which the forces make with
each other, i.e. that </> = TT p*.

235. We may further confirm this conclusion indirectly
by showing that any other supposition as to the relation
between components and resultant is impossible. Let us
* [See Preface.] *

Chap.V.] ILLUSTRATIONS FROM MECHANICS. 363

first assume that a supposition which we wish thus to test
agrees with the foregoing so far as regards the direction of
RI and only makes the length of R exceed or fall short of
the diagonal Z>. Let us suppose then that the first resultant
RI obtained from a and b is greater than the diagonal Z\ of
the parallelogram obtained from a and b with the included
angle & i.e. that R^~ p . D^ where/ is an improper frac-
tion. Now if we combine this R with the force a turned
in the opposite direction, the angle between the two being
TT $ *, the new resultant R^ deduced from them according
to the same supposition must be greater than the diagonal
got from RI and a with this same angle, still greater there-
fore than the other diagonal _Z> 2 which would be got by
combining D^ which is less than R^ and a at the same
angle 77 = </>*. But we know upon purely geometrical
grounds, which are quite independent of all mechanical
assumptions, that this diagonal D is nothing else than the
given force b ; R< 2 then would be greater than ^, whereas we
know for the reasons lately stated that it must be equal to b.
If now once more we compound R^ with the given a at the
angle <J>, the resultant R z which would be thus obtained
must for the same reasons be equal to R l ; but by the pre-
sent supposition it would for the angle f/> be equal to/ times
the diagonal got from /? 2 and a at this angle ; as then R^ is
greater than /, this diagonal also is greater than the diagonal
D got from a and b at the same angle ; supposing it to be
equal to q D l we get R^ = qp . D ly i. e. fi^ is q times as great
as 7?! was. Thus the supposition that the resultant is
greater than the diagonal leads to the absurd conclusion
that it becomes greater and greater every time that we
repeat this manoeuvre in its calculation. The other suppo-
sition that it is smaller than the diagonal, i.e. that/ and q
are vulgar fractions, would lead to an equally impossible
diminution. In order to make this indirect proof complete
it would be necessary to show further that the supposition
* [TT < obviously should be it </> + the angle between A' v and .j

364 THE DISCO VER Y OF GROUNDS OF PROOF. [Book II.

of a resultant of the same length as the diagonal but making
different angles with the given forces would involve a similar
absurdity, viz. that its course would be more and more de-
flected the oftener its calculation was repeated ; and lastly
it would be indispensable to prove that there is no combi-
nation of these suppositions in which the false consequences
of the one would be counteracted by those of the other.
But as the matter stands it is enough to state what the re-
quirements of logic would be ; we may spare ourselves the
trouble of carrying them out at length.

236. Operations of synthesis or combination may always
be carried out to some end, viz. to the result obtained in
each case ; but operations of analysis on the other hand
presuppose an end which we desire to reach, though it is
yet uncertain whether the subject we are treating is produced
by a combination which makes this reverse process of
analysis possible. Even in pure mathematics therefore the
inverse operations lead to difficulties from which the direct
are free ; and similar doubts are suggested by the common
practice of resolving given forces into components, though if
the components were given no doubt would be felt about
combining them. As any force may be split up into count-
less pairs of components, how, it may be asked, are we
entitled to expect that any division which we arbitrarily
choose will have a real validity in the complex tissue of
facts present in the problem before us ? In general terms
this doubt is easily removed. For when we are making
such a resolution in practice we always put one of the com-
ponents in a direction in which some resistance or some
counteracting force is foreseen or known to be present ; we
only resolve therefore for convenience in formulating our
calculation ; what we really do is to compound ; if we com-
bine the given counterforces or resistances JFwith the given
force F y the resultant thus got is identical with that which
would be obtained from the uncancelled remainder of the
onfc component of F and the whole of the other component

Chap.V.] ILLUSTRATIONS FROM MECHANICS. 365

which would meet with no resistance. But a real difficulty
arises when the direction of the resistance itself is not im-
mediately given and an attempt is made in a manner that
seems to me hardly convincing to arrive by an application
of the law of resolution at the principle itself which is here
to be followed. I allude to the supposition that a plane
resists in the direction of its normal only the imparting to it
of a motion which makes with it any angle </>. It is quite
easy to see that this motion may be decomposed into two,
of which one parallel with the plane meets no resistance
because it does not act upon the plane at all, while the
other perpendicular to the plane is annihilated by the resist-
ance of the plane, or at any rate is resisted by it. But how
little right we have to carry out this decomposition here as
one allowed by the nature of the case will appear from the
following considerations.

Let the moving body be a perfectly smooth ball, -and let
it move at an angle <j> against a perfectly smooth plane E
which offers an absolute resistance ; contact then will take
place only in the geometrical point /, to which we must
ascribe the same power of absolute resistance as to all the
other points of E^ however this may be brought about.
Now what all these other points of E have to do with the
result which follows, it is impossible to imagine ; we think
of them indeed when we speak of the plane E \ but as they
are not in contact, they cannot directly contribute anything
to the resistance, and in deducing the result we may set
them entirely aside without altering the conditions on which
the result is to depend. But if we do this and retain the
point/ alone, the proposition about the resistance being at
right angles becomes impossible, because it becomes mean-
ingless ; for to the point p either no line is normal or any
line drawn from it in any direction is normal. But another
principle seems evidently to apply here : surely/, {/"it resists,
will resist in the direction from which comes the motion to
be resisted : there is in the first instance no conceivable

366 THE DISCOVERY OF GROUNDS OF PtfOOF. [Book II.

reason for action in any other direction. If then in our
example / were perfectly fixed, and if at the moment of
contact the line /drawn through the point/ parallel to the
direction of the motion did not pass through the centre of
the ball, p would entirely annihilate the motion of that
thread of the mass which lies in this line /; then for the
rest of the mass of the ball, whose motion would not thus
be annihilated, there would arise a movement of rotation,
which would cause it to turn about the point p. The infer-
ence that the resistance must occur in the direction of the
motion cannot moreover be obviated by conceiving the
moving body to be prismatic in shape, say a cube, of which
one side remains parallel to the plane E while the direction
of its motion makes with E the angle 4>. It is true that in that
case two planes are brought into contact ; but even now
every point of that part of E which is in contact will only
be able to resist the point of the cube's side which it touches
in accordance with the foregoing principle, i.e. in the direc-
tion <j) ; before we could say that it would not be so we
should have to prove that the presence of the adjacent
points q r s of the plane E helps to determine the direction
of the resistance offered by the point / : only this could
render possible in fact that co-operation of the plane which
we have hitherto spoken of, though we have not made use
of it in deducing the result.

And now surely it is clear that we shall never succeed in
proving this so long as we regard E as a geometrical plane
without physical mass and yet with power to offer resistance.
It is not even enough to regard E as the limiting surface of
an inert mass ; we are obliged to add a physical hypothesis
about the forces with which the mass resists encroachment
upon the space it occupies. We must give the plane E
some thickness therefore \ contact will not take place at
one point merely, but the moving body will in /act either
penetrate to a certain depth and then be thrust back by the
resistance of other displaced points of trie mass, or without

Chap.V.] THE TAYLORIAN THEOREM. 367

coming into contact while it is still at a distance it will be
affected by the repulsive forces of the masses united in E.
And then we should have to prove with regard to these
forces of all the points of the mass that in all the other
directions they annihilate one another, but in the direction
of the normal to the limiting surface are added to one
another and combine to make the resistance which annihi-
lates that component of the body's motion which lies in this
normal but in the contrary direction. And indeed it is not
at all surprising that we should be obliged to come back to
an assumption of this kind : motion altogether can only
take place in a real thing, not in a point or a line ; still less
can we hope to calculate resistances without taking count of
that which is alone able to resist, viz. the physical forces of
actual bodies ; surfaces as surfaces and lines as lines always
cut one another without any resistance at all.

1 238. I will add one more mathematical example to
illustrate our general directions about method. The Tay-
lorian theorem attempts to determine the value F(x-\-ti)
which F x^ a function of #, assumes when the variable
quantity x increases from the limiting value which it had in
Fx to the new value x + h. To make the statement as
simple as possible I will subject the problem to certain
limitations : it would take us far too long to enquire here
whether they are superfluous or not. I conceive Fx to be
given* in the shape of an analytical expression which indi-
cates the mathematical operations or relations from which
for every definite value of x flow definite values of Fx ; I
assume that these values of Fx remain finite for every value
of x from o to x -f ^, and that they increase continuously as
x increases continuously between these limits. In pro-
pounding the problem in this form, as one capable of a
universal solution, we directly assume that the growth of

1 I 237,which followed here, is suppressed by desire of the author as
being altogether wrong (' wegen voliigen Irrthums durch den Verfasser
unterdriickt'). See Editor's Preface, and Appendix.

368 THE DISCOVER Y OF GROUNDS OF PROOF. [Book IT.

the function from its value Fx to its new value F(
will follow precisely the same law which the former value
Fx itself followed as x grew from o to its former limiting
value X, and further that this sameness of the generating law
will hold good for each infinitely small increment dh by
which the function now increases precisely as for each in-
finitely small dx by which it formerly increased. From
this it follows that it must be possible to express either
value of the function, and in the first instance to express
Fx, as the sum of an infinite series, each member of which
indicates the increase which takes place as x increases by
the addition of each successive d x. Now if it were the
nature of Fx that for every smallest increase of x, i. e. for
every d x, it increased by the same constant quantity m . dx,
its total value at the end would be the sum of an infinite
series of similar members of the form m dx : the number of
these members would be just as infinite as the number of
d x into which we conceive the final value of x to be divided,
or by the accumulation of which we conceive it to be formed ;
the sum of the series is the integral fm dx~mx, If on the
other hand the increase of Fx for every dx depends upon
the value which the growing x has already attained at the
time when this dx is added, then, if the formula we are
seeking is to hold good for every finite x and //, the series
we now have to take must consist of nothing but similarly
constructed functions of x, relative successively to the con-
tinuously increasing values of x' if we call this function/,*
or/ 1 x, then Fx = < // 1 . d x. Now there is no reason why
we should not repeat with regard to/ 1 * the same consider-
ations which we have already applied toFx-, if* in/ 1 *
now denotes a definite value out of the many values which
x may assume,/ 1 x may also be conceived as the sum of a
series whose infinitely numerous and similarly constructed
members give the increments by which as each dx was
added/ 1 x grew to its limiting value corresponding to that
val*e of x ; and so we get/ 1 x = // 2 x. dx, and generally

Chap. V.] THE TA YLORIAN THEOREM. 369

f m x = yy m +i % . d x. How to obtain from a given function
F x these derivative functions of various grades, f l x, f 2 x>
f m x, and how to work back from the latter to the former,
we may assume to be well known to all who are acquainted
with the infinitesimal calculus.

239. These preliminary remarks really contain the solu-
tion of the problem ; nevertheless I will proceed to trace it
back to the following simple train of thought which may
serve at the same time to illustrate another logical method.

i. Evidently F(x + K) is equal to the sum of its former
value F x and the positive or negative increment JR^ which
F x has received in consequence of the growth of the vari-
able x from x to x -f //. In order to determine the value
of RI we make the simplest supposition, viz. that for each
of these increments dh whose aggregate amounts to h, F x
increases by the same quantity m^ d h ; then m\f dh which
is equal to m . h is the value of R^ or is the total increase
of F x. This m^ is not incalculable. For if, as we through-
out assume, the increase of F x is to depend solely upon
the nature of this function, its given value F x must have
originated in the same manner in which its further growth
is now to take place ; i. e. while x was passing through all
values from o to x the function then in course of formation
must have exhibited for each d x the same increase which
the function thus formed now exhibits for each d h, for
dx differs from d k in name only. Now Fx may be uni-
versally described as the sum of a continuous series, whose
general term is represented by/ 1 x . d x and its last term by
the same expression if x stands for the definite limiting
value which the variable x attains in F x. For each d x
this series increases by/ 1 x . d x ; this quantity/ 1 x must be
constant and be equal to m^ if the growth of Fx up to its
given limiting value is assumed to have taken place in the
same way S the growth from this point up to F(x -\- h).
For every dh therefore F x increases by f 1 x . d h, and the
sum or the integral of these elementary increments, viz.

LOGIC. VOL. I R h

370 THE DISCOVER Y OF GROUNDS OF PROOF. [Book II.

h ( f l x, is the required value of jR r The supposition here
made that f l x is constant and equal to m l may not hold :
but as the general formula must include the cases in which
it does hold good, this second term which we have found
may be accepted as an abiding element of it.

2. Even if this first supposition does not hold yet
F (x + h) is always equal to F x -f- h . f 1 x -f- R^ if we
understand by R^ the positive or negative supplement still
necessary for the complete measurement of the true value
of the function. As this further addition can only be
required because F x does not increase by the same amount
for every dh or d x, i.e. because f 1 x is no constant
quantity, but dependent upon the value which the variable
x has attained at each stage, it is plain that f 1 x in the
second term, hf l x ( R^ of our formula still denotes only
the fixed particular value which the general function f 1 x
now to be conceived as variable, assumes when the variable
x assumes its limiting value x or when the variable h is
equal to o. We cannot therefore retain this second term
h . f 1 x unless to each of the terms/ 1 x . d h of which it is
the sum we add the further increase exhibited by the limit-
ing value of f 1 x contained therein for each increment d h
of the variable h. For this increase again we make the
simplest supposition, viz. that it is the same for each dh
and is equal to m 2 d h. This m. 2 is also capable of determina-
tion. For once more if our supposition is to hold good it
must react upon Fx also \ the same law by which this
function is now to increase must have regulated its origin ;
the increase of f 1 x must have been the same for each d x
and equal to m, 2 d x. Now/ 1 x is the sum of a continuous
series whose general term is/ 2 x . d x ; this then is the very
increment by which this series or its sum f 1 x continuously
increases each time that x is increased by dx\ our con-
dition is fulfilled therefore if we put down/ 2 x as constant
and equal to m tl : then the growth of F x beyond its given
value follows the same law which regulated its formation up

Chap.V.] THE TAYLORIAN THEOREM. 371

to that point. Its total increase therefore is the sum of two
series ; the first of these consists entirely of similar terms
f 1 x . d h, and its sum = ^? x ; the second represented by R^
contains increasing terms, the first termy" 2 x . d h represents
the first new increase which F x exhibits when the former
limiting value x of the variable x is increased by the first
d h, or when the variable /*, growing from o, attains its
first value d h ; each successive (/z+i) fc h term is formed
by adding the same increment f 1 x . d h to the value of the
th term; h .f 2 x . d h therefore is the general term of
this second series, and is what we must add as supplement
to the general term of the first series. The total increase
of F x is therefore the sum of the continuous series

(f l x +hf*x)dh, or h.f l x-\ / 2 -#; the second term

of this expression is the required value of R v

3. If a given function F x were of such a nature that
even this second supposition was not enough to exhaust its
growth, we should still be always able to retain the terms
of the formula already found if we added a fresh jR s to
supplement them. And to determine this Jt 9 we should
repeat the same process as before. We could only require
it because f l x also is not constant, but is dependent upon
the value which x has attained at any point and increases
with tf. Let us assume that these increments are at least
constant for each d h and equal to m^d h. If then we
express f*x as the sum of a continuous series whose
general term is / 3 x . d x, we have but to put down / 3 x as
constant and equal to m at and we thereby make sure that
our general condition is satisfied and that F x has grown to
this its given limiting value in the same way as it is now to
grow beyond it. Now R^ the third term of our formula,
was the sum of a continuous series, whose general term is
h .f 2 x . d k ; if then we form a second series containing the
additions by which jR 2 is to be supplemented, h.f* x .dh
will be the amount by which each (n + 1 ) th term of this

B b 2

372 THE DISCOVER Y OF GROUNDS OF PROOF. [Book II

series exceeds the n* term ; f h ./ 3 x dh therefore or
%*
f 2 x is the general term of this series R y We obtain

the second and third increment of Fx therefore by sum-
ming the continuous series whose general term is now

* 2

and the result is that

! . 2. 3

4. It would be useless to carry this process further \ it
will readily be seen that if we constantly repeat the as-
sumptions here made the required formula will assume the
familiar shape of the Taylorian series, viz.

./* +-- f*x + H * ./ 3 * ......

J 1.2 I . 2. 3 '

But this formula would be of little value if the very assump-
tions on which it rests could not be shown to be the only
admissible assumptions. It would be beyond all doubt
logically correct, but only in the sense in which the
barrenest of tautologies is correct, if it only meant that any
quantity M might always be expressed by a series of quite
arbitrary terms provided that we reserved the right to add a
remaining term R intended to make good all the errors
which we had committed by making M equal to the series.
The formula has a serviceable meaning only when we do
not need this compensating remainder, i.e. when we can
prove that the value of F (x + K) can be completely ex-
pressed either by a finite number of the developed terms, or
by a series of such terms which though infinite yet con-
verges so as to admit of being summed. But how do we
learn that this is the case ? From the fact that for a given
function F x one of its derivative functions f m x turns out
upon actual calculation to be equal to o, that the series

Chap. V.] THE TAYLORIAN THEOREM. 373

therefore breaks off before the term which contains it, *ye
plainly can infer nothing but that there is no further increase
of F x that can be got by the further development of the
series we have taken ; the inference that no other increase
can occur at all would imply that we could prove that this
very mode of calculation must include all increase of which
F x is by its nature capable. Now this point we think no
longer needs special demonstration ; it is contained in the
assumption which we made that F x does not increase
under any other condition than that of the continuous uni-
form increase of x y and that its mathematical structure remains
the same for every one of the values of x which have been
reached. If then a function grows in such a way that for
every d h it exhibits the same constant increase, while at the
same time every d h that thus enters into it becomes the
starting-point of a new constant increase, we get as the
expression of its total increase through the interval h an
infinite series, in whose terms the one set of factors h,

h* h

, depend for their form simply upon this uni-

1.2 1.2 ...m 1 l J r

versal form of growth and are therefore similar in form for
all functions. But in order that this series may give the
specific growth of each particular function in distinction
from that of any other, the other set of factors/ 1 x, f 2 x> / 3 x
are ac^ded to these universal factors in such a way that each
of them indicates the particular magnitude, dependent in
each case upon the nature of the given F x y of the first,
second, third, or m^ increase which occurs for each d h ;
the series, as the complete expression of F (x 4- h), closes
when one of these factors vanishes. The developed terms
of the series above given were therefore not arbitrarily
assumed ; what we meant to do with them was to measure
F (x + h\ not by a standard foreign to the nature of this
function, fcut by the standard supplied by the function itself
and by the nature of its assumed growth ; if by this standard
the value of F (x + h) can be expressed in a finite number

374 TH E DISCOVER Y OF GROUNDS OF PROOF.

of terms or in a number which though infinite admits of
being summed, there can be no increase derived from other
sources which would have to be added to this. For how-
ever a function may grow, provided only that it is subject
at no stage of its growth to the introduction of new con-
ditions from without, the continued repetition of the
assumptions above made (first of a constant increase, then
of a constant positive or negative increase of this increase,
then of a fresh constant positive or negative increase of this
second increase, and so on) will enable us to exhaust the
total value of the resulting growth just as certainly as we are
enabled to express any curved path by properly chosen
epicycles, or any irrational number by an infinite series of
positive and negative powers of ten. Taken in this sense,
as a mere definition of growth, the series remains logically
valid even when it is rendered mathematically useless by
divergence for a demonstrably finite increase of the function.
If it were not so, then, even if it were possible to restore
convergence by transforming the function without altering
its content, the result it yielded could only be regarded as
correct in fact, supposing it could be shown to be correct,
it could not be regarded beforehand as obviously and ne-
cessarily correct : such transformation only serves to bring
within the limits of calculability what holds good as it
stands.

END OF VOLUME I.